Exercises: Continuity. Types

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.

A home delivery app charges a fixed fee depending on the distance from the restaurant. The function defining the delivery cost in pesos, $f(x)$, according to the distance $x$ in kilometers is:

Check whether the function is continuous at the point $x = 3$ by calculating its one-sided limits.

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.

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.

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.

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?

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.

Determine whether the following piecewise function is continuous at the break point $x = 1$:

To do this, check whether the three conditions for continuity are satisfied:

1. Does $f(1)$ exist?

2. Does $\lim_{x \to 1} f(x)$ exist?

3. Do both values match?

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.

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:

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 $x = 0$?

Study the continuity of the following rational function across its domain and classify its points of discontinuity:

For the points where the discontinuity is removable, define a new function $g(x)$ that is continuous at that point.

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:

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?

Determine the values of the parameters $a$ and $b$ so that the following piecewise function is continuous over the whole real line $\mathbb{R}$:

Consider the function $f(x) = \dfrac{|x - 4|}{x - 4}$.

1. a) Calculate the one-sided limits of the function as $x$ tends to $4$.
2. b) State whether the function is continuous at $x = 4$. If it is not, classify the type of discontinuity.
3. c) Sketch the graph of the function in the neighbourhood of the point $x = 4$.

The construction of a bridge has been designed, with its side profile following the function:

1. a) Is the height of the bridge continuous at the junction point $x = 0$?
2. 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 $f(x) = e^{\frac{1}{x-1}}$, analyse its continuity and classify its discontinuities. To do this:

1. Determine the domain of the function.

2. Calculate the one-sided limits at the point where the function is undefined.

3. Interpret the results to indicate whether a vertical asymptote exists and what type of discontinuity it is.

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:

1. a) Determine the values of the parameters $a$ and $b$ so that the fee is continuous over its whole domain.

1. 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:

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:

1. 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.

1. b) Study the continuity of $B(t)$ at $t = 2$ by calculating the one-sided limits. Does the limit exist at that point?

1. 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.

Given the function depending on the real parameters $k$ and $m$:

Find the values of $k$ and $m$ so that $g(x)$ is a continuous function on the set of all real numbers.

Consider the function $f(x) = \frac{x^3 - kx^2 + 3x}{x^2 - 4x + 3}$.

1. a) For which values of $k$ does the function have a removable discontinuity at $x = 3$?
2. b) Having found that value of $k$, redefine the function at that point so that it is continuous at $x = 3$.
3. c) With the value of $k$ obtained, what happens at $x = 1$? Classify the discontinuity at that other point.

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 ($t$):

1. a) Justify whether the level of acceptance has varied continuously over the 10 months.
2. 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 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 $x \to +\infty$.

Let the function be defined by:

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.