Continuity. Types of Discontinuity
A cloud storage plan charges $\$2$ per gigabyte, but past a certain threshold the price changes. In one case the bill rises gradually; in another, once you go over 1 GB, you are suddenly charged $\$40$ more. How can you tell exactly the difference between a rate "with no jumps" and one that does have them, and what is each type of break called? That is continuity.
Continuity at a point: the three conditions
A function is continuous at $x=a$ when its graph does not break right there. The value you approach from both sides is the same value the function takes at that point.
Remember that the limit exists when the one-sided limits exist and are equal. Look at this piecewise function and notice where the graph "joins up":
The functions you already know (polynomials, $e^x$, $\sin x$, roots and logarithms on their domains) are continuous at every point where they are defined. That is why, for a polynomial, the limit at any point is simply evaluating.
Determine whether the following piecewise function is continuous at the break point $x = 1$: $$f(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 3 & \text{if } x = 1 \\…
Determine whether the following piecewise function is continuous at the break point $x = 1$:
$$f(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 3 & \text{if } x = 1 \\ 2x & \text{if } x > 1 \end{cases}$$To do this, check whether the three conditions for continuity are satisfied:
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Does $f(1)$ exist?
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Does $\lim_{x \to 1} f(x)$ exist?
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Do both values match?
Given the function $f(x) = \frac{x^2 - 9}{x - 3}$, state whether the function is defined at $x = 3$. Does the limit $\lim_{x \to 3} f(x)$ exist? Based on your answers,…
Given the function $f(x) = \frac{x^2 - 9}{x - 3}$, state whether the function is defined at $x = 3$. Does the limit $\lim_{x \to 3} f(x)$ exist? Based on your answers, decide whether the function is continuous at that point.
Study the continuity of the polynomial function $f(x) = x^3 - 2x + 5$ at the point $x = 0$. Do you think this function could have any discontinuity at any other real…
Study the continuity of the polynomial function $f(x) = x^3 - 2x + 5$ at the point $x = 0$. Do you think this function could have any discontinuity at any other real number? Explain why.
Reading continuity from a graph
A function is continuous on an interval if you can trace its graph without lifting your pen. Every time you have to lift it, there is a discontinuity. To decide which type it is, ask three questions at the point:
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At what height do the branches arrive from the left and from the right?
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Is there a filled-in point? At what height is it?
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Does the graph go up or down without bound (vertical asymptote)?
Identify which of the following three graphs corresponds to a continuous function across the whole interval shown and explain which visual feature allows you to be sure.
Identify which of the following three graphs corresponds to a continuous function across the whole interval shown and explain which visual feature allows you to be sure.
Observe the following graph representing the Wi-Fi signal level in a shopping center as you move away from the main router. Identify the values of $x$ at which the…
Observe the following graph representing the Wi-Fi signal level in a shopping center as you move away from the main router. Identify the values of $x$ at which the function is not continuous and explain intuitively, with reasoning, why this happens.
Given the following graph of a function $f(x)$ representing the activity level of a mobile application over a time interval, analyze its continuity at the points…
Given the following graph of a function $f(x)$ representing the activity level of a mobile application over a time interval, analyze its continuity at the points $x = -2$, $x = 1$ and $x = 3$. State in each case the type of discontinuity (removable, finite jump, or infinite jump), justifying your answer by examining the one-sided limits and the value of the function.
Classification of discontinuities
When one of the three conditions fails, there are three ways it can fail. Each has its own name and its own look on a graph.
Classify the type of discontinuity presented by the function shown in the image at the point $x = 2$. Justify whether it is a removable discontinuity, a finite jump…
Classify the type of discontinuity presented by the function shown in the image at the point $x = 2$. Justify whether it is a removable discontinuity, a finite jump discontinuity, or an infinite jump discontinuity.
Consider the function $f(x) = \dfrac{|x - 4|}{x - 4}$. a) Calculate the one-sided limits of the function as $x$ tends to $4$. b) State whether the function is continuous…
Consider the function $f(x) = \dfrac{|x - 4|}{x - 4}$.
- a) Calculate the one-sided limits of the function as $x$ tends to $4$.
- b) State whether the function is continuous at $x = 4$. If it is not, classify the type of discontinuity.
- c) Sketch the graph of the function in the neighbourhood of the point $x = 4$.
Let the function be defined by: $$f(x) = \begin{cases} \frac{1}{1 + e^{1/x}} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$$ Study the continuity of the…
Let the function be defined by:
$$f(x) = \begin{cases} \frac{1}{1 + e^{1/x}} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$$Study the continuity of the function at the point $x = 0$ by analysing its one-sided limits. Is it continuous from the right? And from the left? Classify, with reasoning, the type of discontinuity it presents.
Piecewise functions: finding parameters
Many rates and costs are defined in pieces: electricity, the Metrobús, a data plan. For the cost not to jump when switching from one piece to the next, the expressions have to be "joined" properly. This translates into one equation at each joining point.
Move $k$ and $m$ until the two branches touch at $x=0$. What value must $m$ have?
Determine the values of the parameters $a$ and $b$ so that the following piecewise function is continuous over the whole real line $\mathbb{R}$: $$f(x) = \begin{cases} x…
Determine the values of the parameters $a$ and $b$ so that the following piecewise function is continuous over the whole real line $\mathbb{R}$:
$$f(x) = \begin{cases} x + 2a & \text{if } x < -1 \\ 3x + b & \text{if } -1 \leq x < 2 \\ x^2 - 5 & \text{if } x \geq 2 \end{cases}$$The construction of a bridge has been designed, with its side profile following the function: $$h(x) = \begin{cases} \sqrt{x+4} & \text{if } -4 \leq x < 0 \\ \dfrac{x^2…
The construction of a bridge has been designed, with its side profile following the function:
$$h(x) = \begin{cases} \sqrt{x+4} & \text{if } -4 \leq x < 0 \\ \dfrac{x^2 - 4}{x - 2} & \text{if } 0 \leq x < 4 \end{cases}$$- a) Is the height of the bridge continuous at the junction point $x = 0$?
- b) Is there any point of discontinuity in the interval $[0, 4)$? If so, explain whether it poses a real construction problem (removable discontinuity) or if it represents a break in the profile (jump).
Given the function depending on the real parameters $k$ and $m$: $$g(x) = \begin{cases} \frac{k \times \text{sen}(x)}{x} & \text{if } x < 0 \\ x^2 + m & \text{if } 0 \le…
Given the function depending on the real parameters $k$ and $m$:
$$g(x) = \begin{cases} \frac{k \times \text{sen}(x)}{x} & \text{if } x < 0 \\ x^2 + m & \text{if } 0 \le x \le 1 \\ \frac{\sqrt{x} - 1}{x - 1} & \text{if } x > 1 \end{cases}$$Find the values of $k$ and $m$ so that $g(x)$ is a continuous function on the set of all real numbers.
Rational functions: domain and type of discontinuity
In a rational function the only candidates for discontinuity are the points that make the denominator zero. Factoring the numerator and denominator decides, at each one, whether the discontinuity is removable or a vertical asymptote.
For long-term modeling (for example, the population of a bacteria colony as $t$ grows) you also need the limit at infinity of a rational function. If the numerator and denominator have the same degree, the limit is the quotient of the leading coefficients: $\displaystyle\lim_{t\to\infty}\dfrac{at-5}{t-2}=a$.
A content creator analyses the growth of their followers using a function that has suffered a server error. The function is $g(x) = \frac{x-1}{x^2-1}$. State for which…
A content creator analyses the growth of their followers using a function that has suffered a server error. The function is $g(x) = \frac{x-1}{x^2-1}$. State for which values of $x$ the function is not continuous. What happens in the denominator that causes continuity to fail?
Study the continuity of the following rational function across its domain and classify its points of discontinuity: $$f(x) = \frac{x^2 - x - 6}{x^2 - 9}$$ For the points…
Study the continuity of the following rational function across its domain and classify its points of discontinuity:
$$f(x) = \frac{x^2 - x - 6}{x^2 - 9}$$For the points where the discontinuity is removable, define a new function $g(x)$ that is continuous at that point.
Consider the function $f(x) = \frac{x^3 - kx^2 + 3x}{x^2 - 4x + 3}$. a) For which values of $k$ does the function have a removable discontinuity at $x = 3$? b) Having…
Consider the function $f(x) = \frac{x^3 - kx^2 + 3x}{x^2 - 4x + 3}$.
- a) For which values of $k$ does the function have a removable discontinuity at $x = 3$?
- b) Having found that value of $k$, redefine the function at that point so that it is continuous at $x = 3$.
- c) With the value of $k$ obtained, what happens at $x = 1$? Classify the discontinuity at that other point.
Real-world models: rates, costs, and profits
These exercises combine everything above. The procedure is the same: identify the "suspicious" points (joining points and zeros of the denominator), compute one-sided limits, and compare with the value of the function. Then interpret the result.
An express delivery company sets its monthly rates for corporate clients according to the volume of packages sent ($x$, in hundreds of units). The total cost function…
An express delivery company sets its monthly rates for corporate clients according to the volume of packages sent ($x$, in hundreds of units). The total cost function ($C(x)$, in thousands of pesos) is given by:
$$C(x) = \begin{cases} \dfrac{x^2 + 2x}{x} & \text{if } 0 < x < 2 \\ k \times (x - 1) & \text{if } x \geq 2 \end{cases}$$Calculate the value of the parameter $k$ so that the function is continuous at $x = 2$. What type of discontinuity does the function present at the point $x = 0$?
The growth of a bacterial colony in a controlled environment follows a pattern that changes when a nutrient runs out 3 hours after starting the experiment. The…
The growth of a bacterial colony in a controlled environment follows a pattern that changes when a nutrient runs out 3 hours after starting the experiment. The population $P(t)$ (in millions) is modelled by the function:
$$P(t) = \begin{cases} t^2 + 1 & \text{if } 0 \leq t < 3 \\ \dfrac{at - 5}{t - 2} & \text{if } t \geq 3 \end{cases}$$Where $t$ is the time in hours. Determine the value of $a$ so that the population growth is continuous at the instant the diet changes ($t = 3$). In the long term ($t \to \infty$), what value does the population tend towards if this second pattern continues?
Given the function $f(x) = e^{\frac{1}{x-1}}$, analyse its continuity and classify its discontinuities. To do this: 1. Determine the domain of the function. 2. Calculate…
Given the function $f(x) = e^{\frac{1}{x-1}}$, analyse its continuity and classify its discontinuities. To do this:
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Determine the domain of the function.
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Calculate the one-sided limits at the point where the function is undefined.
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Interpret the results to indicate whether a vertical asymptote exists and what type of discontinuity it is.
Back to the beginning: a rate that "rises gradually" is continuous: the one-sided limits match the value at each threshold. The one that goes up $\$40$ all at once past 1 GB has a finite jump discontinuity at that threshold, and the one-sided limits prove it.
Once you have mastered the above, try the most challenging ones:
A cloud services company offers a storage plan whose monthly fee $C(x)$, in pesos, depends on the gigabytes used $x$. The function is given by the following expression:…
A cloud services company offers a storage plan whose monthly fee $C(x)$, in pesos, depends on the gigabytes used $x$. The function is given by the following expression:
$$C(x) = \begin{cases} 2x + a & \text{if } 0 \le x \le 10 \\ \frac{x^2 - 100}{x - 10} + b & \text{if } 10 < x < 50 \\ \ln(x - 49) + 45 & \text{if } x \ge 50 \end{cases}$$- a) Determine the values of the parameters $a$ and $b$ so that the fee is continuous over its whole domain.
- b) If $a$ has the value found in the previous part but $b$ were different from the value found, what type of discontinuity would the function have at $x = 10$? Justify your answer by calculating the one-sided limits.
Study the continuity of the following piecewise function and classify all its points of discontinuity: $$f(x) = \begin{cases} \frac{e^x}{x} & \text{if } x < 0 \\…
Study the continuity of the following piecewise function and classify all its points of discontinuity:
$$f(x) = \begin{cases} \frac{e^x}{x} & \text{if } x < 0 \\ \frac{x^2 - 1}{x^2 - 3x + 2} & \text{if } 0 \le x < 2 \\ \frac{3}{x - 2} & \text{if } x > 2 \end{cases}$$State, with reasons, whether any of the discontinuities found are removable.
A financial analyst models the expected profit $B(t)$, in thousands of pesos, of a technology investment over time $t$ (in years) using the function: $$B(t) = \frac{|t^2…
A financial analyst models the expected profit $B(t)$, in thousands of pesos, of a technology investment over time $t$ (in years) using the function:
$$B(t) = \frac{|t^2 - 4|}{t - 2}$$- a) Draw the graph of the function on the interval $[0, 5]$. To do so, write $B(t)$ as a piecewise-defined function (without absolute value) and build a table of values.
- b) Study the continuity of $B(t)$ at $t = 2$ by calculating the one-sided limits. Does the limit exist at that point?
- c) Classify the type of discontinuity, calculate the size of the jump (in thousands of pesos) and explain its meaning in the context of the company's profit.
You can do it on paper and send a photo with your answer.
A sociological study establishes that the level of acceptance of a new law, $L(t)$, on a scale from 0 to 100, follows the following model during the first 10 months…
A sociological study establishes that the level of acceptance of a new law, $L(t)$, on a scale from 0 to 100, follows the following model during the first 10 months ($t$):
$$L(t) = \begin{cases} t^2 - 4t + 50 & \text{if } 0 \le t \le 4 \\ \frac{50t - 100}{t - 2} & \text{if } 4 < t \le 10 \end{cases}$$- a) Justify whether the level of acceptance has varied continuously over the 10 months.
- b) If a change in the communication campaign occurred in month $t = 4$, was there a sudden jump in acceptance? If so, calculate the size of the jump and classify the discontinuity.
Consider the function $f(x) = \frac{x^2 - a}{x^3 + bx + c}$. It is known that the function has an infinite discontinuity at $x = 1$ and $x = -3$. In addition, it has a…
Consider the function $f(x) = \frac{x^2 - a}{x^3 + bx + c}$. It is known that the function has an infinite discontinuity at $x = 1$ and $x = -3$. In addition, it has a removable discontinuity at $x = 2$.
Find the values of the parameters $a, b$, and $c$. With these values, calculate the limit of the function as $x \to +\infty$.