Exercises: Continuous functions. Discontinuities

Observe the following graph of the function $f(x)$ and determine at which points of its domain it has a discontinuity. Classify each of them, stating whether it is a removable discontinuity, a finite jump discontinuity or an infinite discontinuity.

Given the rational function $f(x) = \frac{3x-1}{x+5}$, state its domain of definition and the point at which it has a discontinuity. What type of discontinuity is it? Justify your answer by studying the behaviour of the function around that point.

A piecewise function is given by the following expression:

$f(x) = \begin{cases} 2x + 3 & \text{if } x < 1 \\ 4 & \text{if } x \ge 1 \end{cases}$

Calculate the value of the function at $x=1$ and the one-sided limits as $x$ approaches 1. Is the function continuous at that point? If not, classify the discontinuity.

In a city bicycle car park, the charge is 2 € for the first hour or fraction thereof, and 1.50 € for each additional hour or fraction thereof. If we plot the function relating continuous parking time (in hours) to the price to pay (in euros), what type of discontinuity will the graph show at integer values of time ($t=1, 2, 3\dots$)?

Consider the function $f(x) = \frac{x^2-16}{x-4}$.

Calculate the value of the function at $x=4$ if possible. Next, simplify the algebraic expression and calculate the limit as $x$ approaches 4. What type of discontinuity does the function show at $x=4$?

Find the points of discontinuity of the function $f(x) = \frac{x+2}{x^2-3x-10}$ and state what type each one is, analysing whether or not the numerator is zero at those points.

Study the continuity at $x=0$ of the following function:

$f(x) = \begin{cases} -x^2 & \text{if } x \le 0 \\ \frac{1}{x} & \text{if } x > 0 \end{cases}$

Classify the existing discontinuity at that point.

A farm water tank fills at a constant rate for 10 minutes until it reaches 500 litres. At that exact moment, an emergency quick-drain valve opens, emptying the tank completely almost instantaneously. If we draw the graph of the volume of water as a function of time, what type of mathematical discontinuity is observed at minute 10?

Correos de España applies different rates for national shipping of standard letters and postcards depending on the envelope's weight $x$ (in grams): for weights up to $20\text{ g}$, postage costs $0,85\text{ €}$; for weights over $20\text{ g}$ and up to $50\text{ g}$, it costs $0,95\text{ €}$; and for weights over $50\text{ g}$ and up to $100\text{ g}$, it costs $1,40\text{ €}$.

1. a) Write the piecewise function $P(x)$ that determines the shipping price for any weight in the interval $(0, 100]$.
2. b) Study analytically the continuity of the function at $x = 20\text{ g}$. To do this, calculate $P(20)$, the limit from the left $\lim_{x \to 20^-} P(x)$, and the limit from the right $\lim_{x \to 20^+} P(x)$.
3. c) Classify the type of discontinuity the function has at $x = 20\text{ g}$ and explain what practical impact it has on two people who drop off letters whose contents weigh exactly $20\text{ g}$ and $20,1\text{ g}$.

Lucas has to analyze the continuity of the function $f(x) = \dfrac{x^2 - 9}{x - 3}$ at the point $x = 3$. On his exam, he writes: "Since the denominator vanishes at $x = 3$, one cannot divide by zero; therefore, the function necessarily has a vertical asymptote at that point and presents an infinite discontinuity".\n\nIdentify the conceptual mistake Lucas makes, explain what preliminary check he omitted in the numerator, calculate the limit of the function as $x$ approaches $3$ by factoring the expression, and correctly classify the type of discontinuity that $f(x)$ has at $x = 3$.

Calculate the value that parameter $k$ must have for the following function to be continuous throughout its domain:

$f(x) = \begin{cases} x^2 - 3x + k & \text{if } x \le 2 \\ 5x - 7 & \text{if } x > 2 \end{cases}$

An electricity company offers a tariff for SMEs: a fixed 15 € plus 0.12 € per kWh consumed, up to a maximum of 100 kWh. If consumption exceeds 100 kWh, the tariff changes and ALL consumed kWh are charged at 0.10 € (removing the fixed cost of 15 €).

Write the piecewise function determining the total cost as a function of the consumed kWh. Is the function continuous at an exact consumption of 100 kWh? If not, state the type of discontinuity and the "jump" in price.

Observe the graph of the function $g(x)$ and answer the following questions:

1. Determine the domain of the function.

2. Identify the points of discontinuity and classify them.

3. Write the value of the one-sided limits at $x = -1$ and at $x = 3$.

Given the function $f(x) = \frac{x^2-x-2}{x^2-1}$, determine its domain of definition. Analytically study the type of discontinuity it presents at $x=1$ and at $x=-1$ by calculating the corresponding one-sided limits.

A 100-metre sprinter accelerates uniformly from rest until reaching a speed of 8 m/s in 4 seconds. They maintain that constant speed for 10 seconds. Right at that moment, they pull a muscle and their speed instantaneously drops to 0 m/s.

Express the speed function $v(t)$ piecewise with respect to time $t$ and classify the discontinuity that occurs at the moment of the injury.

Study the continuity of the function $f(x) = \frac{3}{x-2} + \frac{x}{x+1}$. Identify the points where it is not continuous and analytically justify what type of discontinuity it is in each case.

Determine the value of $a$ so that the function $f(x) = \frac{x^2+ax-10}{x-2}$ has a removable discontinuity at $x=2$. What would be the value of the limit of the function as $x$ approaches 2 in that specific case?

Consider the function defined by $f(x) = \frac{|x+3|}{x+3}$.

Rewrite the function as a piecewise function (without using the modulus). Next, study its continuity at $x=-3$ and classify the discontinuity.

A technical team needs to design an analytical function $f(x)$ to model the calibration of a thermal sensor that exhibits singular behaviors at two key temperatures:

1. At $x = -2$, the sensor outputs a point reading set at $f(-2) = 1$, but the trend of the readings from both the left and the right converges towards the value $-5$ (removable discontinuity).

2. At $x = 3$, the value of the readings shoots up uncontrollably towards infinity as it approaches that point (infinite discontinuity).

3. For any other real value of $x$, the sensor responds continuously.

Construct the algebraic expression of a function (you can use a piecewise function or a rational expression) that satisfies these three conditions simultaneously. Justify your proposal by checking the value of the function and the one-sided limits at $x = -2$ and at $x = 3$.

A municipality is debating two alternative models to bill domestic water supply based on the monthly volume consumed $x$ (in $\text{m}^3$):

• Model A (continuous bracket): if you consume up to $10\text{ m}^3$, you pay a fixed fee of $12\text{ €}$. If you exceed $10\text{ m}^3$, you pay those $12\text{ €}$ plus $1,50\text{ €}$ for each cubic meter that exceeds $10\text{ m}^3$.

• Model B (abrupt block): if you consume up to $10\text{ m}^3$, you pay a flat rate of $12\text{ €}$. However, if you consume more than $10\text{ m}^3$, you are billed for the entire consumed volume at a rate of $1,80\text{ €}/\text{m}^3$ from the very first cubic meter.

1. a) Write the formula of the piecewise function that defines the monthly cost in each model: $C_A(x)$ and $C_B(x)$.
2. b) Before operating formally, estimate which of the two models will generate complaints from residents for causing a disproportionate increase between two almost identical consumptions of $9,9\text{ m}^3$ and $10,1\text{ m}^3$.
3. c) Analytically verify your estimate by studying the continuity of $C_A(x)$ and $C_B(x)$ at the critical point $x = 10\text{ m}^3$ by calculating their one-sided limits and their values at that point. Classify the discontinuity that appears and conclude which of the two models is more equitable.

Find the values of the parameters $a$ and $b$ for the function $f(x)$ to be continuous throughout the set of real numbers:

$f(x) = \begin{cases} \frac{a}{x} & \text{if } x < -1 \\ x^2 + b & \text{if } -1 \le x \le 2 \\ 3x - 1 & \text{if } x > 2 \end{cases}$

In a fictitious country, annual income tax is calculated in the following progressive manner:

• Incomes up to 20 000 € pay 10% on the total.

• The tax bracket exceeding 20 000 € and up to 50 000 € pays 20% (added to what was already paid on the first 20 000 €).

• The tax bracket exceeding 50 000 € pays 30% (added to the previous amount).

Write the piecewise function $T(x)$ that calculates the total tax payable for an income $x$. Analytically show that the function $T(x)$ is mathematically continuous at $x=20\,000$ and at $x=50\,000$.

Given the rational function $f(x) = \frac{x^3 - 3x^2 - x + 3}{x^2 - 4x + 3}$, factorise the polynomials in the numerator and the denominator using Ruffini's rule and solving equations to find the domain of the function. Study and classify in detail the discontinuities it has at the points excluded from the domain.

The rate of a chemical reaction $R(T)$ as a function of temperature $T$ (in °C) is given by a piecewise mathematical model. For temperatures up to the boiling point ($T \le 100$), $R(T) = 3T + k$. For higher temperatures ($T > 100$), the model changes to $R(T) = a \cdot 2^{T-100}$.

It is known that at the boiling point ($T=100$) a finite jump discontinuity occurs, such that the rate suddenly increases by 50 units compared to the left-sided limit. If the rate right at the moment of boiling ($T=100$) is 320 units, calculate the exact values of parameters $k$ and $a$.

Study the continuity of the function $f(x) = \frac{\sqrt{x+9}-3}{x}$ at the point $x=0$. To do this, multiply the numerator and denominator by the conjugate expression of the numerator, calculate the limit as $x$ approaches 0, and classify the resulting discontinuity.

Find the values of $m$ and $n$ so that the rational function $f(x) = \frac{mx^2 + nx - 6}{x^2 - 4}$ has a removable discontinuity at $x=2$ and an infinite discontinuity at $x=-2$.

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

Write it as a piecewise function, rigorously eliminating the modulus. Study its continuity at $x=2$ by calculating the one-sided limits and classify the discontinuity.

An underwater exploration drone of mass $m$ moves through a viscous medium. Its vertical velocity $v(t)$ experiences an infinite discontinuity at time $t=5$ due to an extreme current, and a removable discontinuity at $t=2$ due to a transient measurement sensor fault.

Propose a rational algebraic expression for $v(t)$ that simultaneously satisfies these two conditions and, furthermore, starts from rest at time zero ($v(0)=0$). Justify your answer by analysing the limits of the function you have proposed.

Given the piecewise function:

$f(x) = \begin{cases} \dfrac{x^2 + 2x - 3}{x - 1} & \text{if } x < 1 \\ 4 & \text{if } x = 1 \\ 3x + 1 & \text{if } x > 1 \end{cases}$

To determine whether $f(x)$ is continuous at $x = 1$, two students propose different approaches:

• Method 1 (Alba's numerical approximation): calculate the value of $f(x)$ at several numbers very close to $1$, such as $x = 0,99$, $x = 0,999$, $x = 1,001$, and $x = 1,01$, and visually observe whether the values approach $f(1) = 4$.

• Method 2 (Bruno's analytical resolution): simplify the indeterminacy of the left branch by factoring the polynomial, algebraically calculate $\lim_{x \to 1^-} f(x)$ and $\lim_{x \to 1^+} f(x)$, and check whether both match $f(1)$.

1. a) Apply Method 2 step by step: calculate the one-sided limits, compare them with $f(1)$, and formally classify the continuity or discontinuity of the function at $x = 1$.
2. b) Critically compare both methods: explain why Method 1 does not guarantee rigorous mathematical certainty even though it is intuitively useful, and justify in what real-world situation (for example, in experimental data collection where no algebraic formula exists) Method 1 would be the only viable alternative.

In Spain, the Special Tax on Certain Means of Transport (registration tax) levies the official $\text{CO}_2$ emissions of each new vehicle according to the brackets established by tax regulations:

• Bracket 0: emissions $x \le 120\text{ g/km} \rightarrow 0\,\%$ on the tax base.

• Bracket 1: emissions $120 < x < 160\text{ g/km} \rightarrow 4,75\,\%$ on the tax base.

• Bracket 2: emissions $160 \le x < 200\text{ g/km} \rightarrow 9,75\,\%$ on the tax base.

• Bracket 3: emissions $x \ge 200\text{ g/km} \rightarrow 14,75\,\%$ on the tax base.

A car has a tax base (price of the vehicle before taxes) of $24.000\text{ €}$.

1. a) Express the function $I(x)$ that determines the amount of the tax in euros based on the emissions $x$ for the interval $[100, 180]\text{ g/km}$.
2. b) Study with justification the continuity of $I(x)$ at the bracket boundaries $x = 120$ and $x = 160$. Calculate the one-sided limits and the value of the function at each boundary, and classify the discontinuities found, indicating the exact magnitude of each jump in euros.
3. c) A car manufacturer manages to reduce, through aerodynamic improvements, the approved emissions of a vehicle from $120,4\text{ g/km}$ to $119,7\text{ g/km}$. Relying on your mathematical results, quantify the tax savings resulting from that reduction of only $0,7\text{ g/km}$ and explain why in economics and taxation this phenomenon is known as the "cliff edge effect" (or "threshold effect"). Relate this concept to jump discontinuities.