Exercises: Continuity. Types

Observe the following graph representing the Wi-Fi signal level in a shopping centre 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 euros, $f(x)$, according to the distance $x$ in kilometres 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?

Lucas states in class: "To study the continuity of $f(x) = \frac{x^2 - 9}{x - 3}$ at $x = 3$, I first look at the denominator. Since it vanishes ($3 - 3 = 0$), the function is not defined at that point and, therefore, there must necessarily be a vertical asymptote, that is, an infinite jump discontinuity." Identify the reasoning error Lucas makes, explain which essential condition of the definition of continuity he omitted when classifying the discontinuity, and determine the actual type of discontinuity that $f$ has at $x = 3$, justifying it with the calculation of limits.

In a short-stay car park in a Spanish city, the fee $C(t)$ (in euros) charged according to the parked time $t$ (in hours) is given by the expression:

1. Before performing any algebraic calculations, estimate whether the change in pricing at $2$ hours will produce a sharp jump in the amount payable or if the function will join smoothly (continuously). Justify your prediction thinking about the commercial logic of charging for time used.

2. Check your estimate analytically by applying the three continuity conditions at $t = 2$: calculate $\lim_{t\to 2^-} C(t)$, $\lim_{t\to 2^+} C(t)$, and $C(2)$. From the results, conclude with reasoning whether your prediction was correct and classify the behavior of the function at that point.

Given the following graph of a function $f(x)$ representing the activity level of a mobile application over a time interval, analyse 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 euros) 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.

Design the analytical expression of a piecewise function $f(x)$, with domain on all of $\mathbb{R}$, that simultaneously satisfies the following three conditions:

1. It has a removable discontinuity at $x = -2$, but the function is defined at that point with $f(-2) = 5$.

2. It has a jump (finite) discontinuity at $x = 1$, with a jump of exactly $3$ units (that is, $|\lim_{x\to 1^+} f(x) - \lim_{x\to 1^-} f(x)| = 3$).

3. It is continuous at any other point of $\mathbb{R}$.

Show the complete formula of your function and justify analytically, by calculating the corresponding one-sided limits and checking the continuity conditions, that your proposal rigorously satisfies each of the requirements.

An irrigation community applies a monthly rate $P(v)$ (in euros) for the volume of water $v$ (in cubic meters, $\text{m}^3$) supplied to an agricultural holding, according to the following piecewise function:

where $15\text{ \euro}$ is the fixed service charge and $k$ is a constant adjustment term.

1. a) Determine the value of $k$ so that the billing function is continuous at $v = 100\text{ m}^3$, preventing an irrigator from experiencing an unjustified jump in total cost for slightly exceeding the threshold of $100\text{ m}^3$.
2. b) If due to a management error $k = -10$ were set, analyze continuity at $v = 100\text{ m}^3$, calculate the magnitude of the jump, and interpret economically who is harmed by this discrepancy (whether the farmer or the irrigation community).

A cloud services company offers a storage plan whose monthly fee $C(x)$, in euros, 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 euros, 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 euros) 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.

To study the continuity of the function $g(x) = \frac{|x - 1|}{x^2 - 3x + 2}$ at the points $x = 1$ and $x = 2$, two students propose different approaches:

• Strategy A (Carla): First express the function in two branches without absolute value according to the sign of $(x - 1)$, simplify the fraction algebraically in each piece by factoring the denominator, and then evaluate the one-sided limits and the existence of the function at each problematic point.

• Strategy B (Marcos): Leave the expression with the absolute value intact and directly calculate the one-sided limits at $x = 1$ and $x = 2$ by mentally substituting very close numerical values (for example, $1,01$ and $0,99$) to deduce the sign of the numerator and the denominator.

1. a) Apply Carla's strategy to analyze and classify in detail the type of discontinuity at $x = 1$.
2. b) Compare both procedures for the analysis of the point $x = 2$. Why can Marcos's strategy lead to serious errors or unjustified indeterminate forms in this type of rational functions with absolute value? Explain which method offers greater deductive solidity.
3. c) Finally, classify the discontinuity at $x = 2$ using the method you have justified as the safest.

In a tax reform proposal to tax the net profit $B$ (in thousands of euros) of SMEs, the Ministry of Finance proposes a tax amount $T(B)$ (in thousands of euros) structured in three brackets:

To avoid the so-called "cliff effect" or jump error (where a marginal increase in profit causes the company to suddenly pay a disproportionately higher tax), the function $T(B)$ must be continuous over its entire domain $[0, +\infty)$. In addition, according to revenue criteria, a company declaring a profit of $40\text{ thousand euros}$ must pay exactly $7\text{ thousand euros}$ in taxes.

1. a) Set up the system of equations necessary to guarantee the continuity of the function at $B = 20$ and at $B = 60$, incorporating the condition on the tax amount for $B = 40$.
2. b) Solve the system and find the values of the parameters $m$, $n$, and $p$ with reasoning.
3. c) If due to a typo in the publication of the law $p = -4$ were set, analyze the continuity at $B = 60$, classify the resulting discontinuity by calculating the size of the jump, and explain why this error would disincentivize a company with a profit of $60\text{ thousand euros}$ from trying to earn an additional $100\text{ euros}$.

Topic challenge. Where is youth digital consumption heading? Investigate the patterns in your environment by comparing them with INE data and model their long-term behaviour.

1. Sampling and counting: Check on the INE website the average daily time that young people spend on content platforms. Conduct a micro-survey with $15$ classmates recording their hours of use. How many different samples of $4$ people could you select to compare profiles? Apply combinatorics ($C_{15,4}$).

2. Hypothesis testing: Formulate a two-tailed hypothesis test with a significance level $\alpha = 0,05$ to check whether the mean time in your environment differs significantly from the national INE figure.

3. Modelling and limits: Define a piecewise function $f(t)$ that models cost or consumption as a function of time $t$, with a tariff change at $2$ hours. Calculate the one-sided limits at $t = 2$ to check its continuity. Then, evaluate $\lim_{t \to +\infty} f(t)$ by resolving the indeterminate form $\left[\frac{\infty}{\infty}\right]$ to predict whether consumption tends to stabilise or grows indefinitely.

4. Final product: Prepare a one-page executive report (using GeoGebra or a spreadsheet) that includes the data, the formulated test, the analytical study of limits, and your justified conclusion.

Assessment criteria:

• Precision in the combinatorial count and in the development of the hypothesis test.

• Algebraic rigour in the calculation of one-sided limits, limits at infinity, and resolution of indeterminate forms.

• Effective use of digital tools and critical coherence in the drawn conclusions.