Exercises: Continuity of functions

Study the continuity of the function $f(x) = \frac{x^2 - 16}{x - 4}$. State the points at which it is discontinuous and classify the type of discontinuity present.

Determine the domain of the function $g(x) = \sqrt{x^2 - 25}$ and justify in which intervals the function is continuous.

Look at the following graphical representation of a function $f(x)$ and determine the points where it has discontinuities, classifying them as removable, finite jump, or infinite jump discontinuities.

Express the function $h(x) = |x - 3|$ as a piecewise-defined function and prove analytically that it is continuous at $x = 3$.

Given the exponential function $g(x) = e^{\frac{1}{x-1}}$, study its continuity at the point $x = 1$ by calculating the one-sided limits.

Study the continuity of the logarithmic function $f(x) = \ln(8 - 2x)$, clearly stating the interval of real numbers where it is continuous.

The rational function $f(x) = \frac{x^2 + 2x - 8}{x - 2}$ is not defined at $x = 2$. What numerical value should $f(2)$ take so that the function is continuous at that point?

Study the continuity of the trigonometric function $g(x) = \frac{1}{\cos x}$ in the open interval $(0, 2\pi)$. At which points does it have discontinuities and what types are they?

Lucía tries to prove that the function $f(x) = \frac{1}{x - 2}$ has at least one root in the interval $[1, 3]$ by applying Bolzano's Theorem. Her reasoning is as follows: «I calculate the images of the endpoints: $f(1) = \frac{1}{1 - 2} = -1 < 0$ and $f(3) = \frac{1}{3 - 2} = 1 > 0$. Since the signs are opposite ($f(1) \cdot f(3) < 0$), by Bolzano's Theorem there necessarily exists a value $c \in (1, 3)$ such that $f(c) = 0$».

1. a) Point out precisely the conceptual error Lucía made when checking the hypotheses of the theorem.
2. b) Study the continuity of $f(x)$ on the interval $[1, 3]$ and classify the discontinuity it has in said interval.
3. c) Solve the equation $f(x) = 0$ in $\mathbb{R}$ and explain why the curve goes from negative to positive values without ever crossing the x-axis.

Consider the function $f(x) = \frac{2x^2 - 5x - 3}{x - 3}$ for all $x \ne 3$, which is assigned the specific value $f(3) = 5$.

1. a) Before carrying out algebraic operations, make a numerical estimate of the value that $f(x)$ approaches as $x$ gets close to $3$: evaluate the expression at $x = 2,99$ and at $x = 3,01$. In light of these two values, what do you estimate is the value of $\lim_{x \to 3} f(x)$?
2. b) Calculate $\lim_{x \to 3} f(x)$ analytically by factoring the numerator, and check whether your previous estimate was correct.
3. c) Using the limit obtained and the defined value $f(3) = 5$, determine whether $f$ is continuous at $x = 3$. If it is not, classify the type of discontinuity and state what value should have been assigned to $f(3)$ for the function to be continuous on all of $\mathbb{R}$.

The charge for a public car park is calculated using the function $C(t) = 2 + 1,5 \cdot \text{ENT}(t)$, where $t$ is the time in hours and $\text{ENT}(t)$ represents the floor function of $t$.

Identify the points of discontinuity of this function in the interval $(0, 4)$ and explain the practical meaning of these discontinuities for a car park user.

You can do it on paper and send a photo with your answer.

Using Bolzano's Theorem, show that the equation $x^3 + 2x - 4 = 0$ has at least one real solution in the interval $(1, 2)$. Justify all the conditions necessary to apply the theorem.

The temperature of an industrial oven in degrees Celsius, $T(t)$, as a function of time in minutes $t$, is given by:

$T(t) = \begin{cases} 20 + 15t & \text{if } 0 \le t < 10 \\ k \cdot 0,9^{t-10} & \text{if } t \ge 10 \end{cases}$

Determine the value of $k$ so that there is no sudden jump in temperature at 10 minutes. What physical phenomenon does the function represent from $t = 10$ onwards?

You can do it on paper and send a photo with your answer.

The following graph illustrates the principle of Bolzano's Theorem for a continuous polynomial function $f(x) = x^3 - 4x - 2$. By observing the graph and evaluating the function, find an interval of length 1 (that is, $[n, n+1]$ with $n \in \mathbb{Z}$) where the existence of a real root can be guaranteed, justifying your answer.

A mobile phone company offers a data tariff with the following conditions: a fixed fee of $10$ € is paid, which includes the first $5$ GB. For each additional GB up to $20$ GB, $2$ € is charged. Beyond $20$ GB, the price follows the expression $a + 1,5x$, where $x$ is the total GB consumed.

Find the value of the parameter $a$ so that the function representing the total cost is continuous at all times, and write the complete algebraic expression for this function.

Study the continuity of the rational function $f(x) = \frac{x^3 - 3x^2 - x + 3}{x^2 - x - 6}$ in detail. Classify the types of discontinuity it has at the points where it is not defined.

Given the function $f(x) = \frac{x^2 + kx - 6}{x - 2}$, calculate the value of the parameter $k$ so that the discontinuity at $x = 2$ is removable. Once found, what will be the value of the limit of the function at that point?

Given the function with modulus $f(x) = \frac{x^2-9}{|x-3|}$, rewrite it as a piecewise-defined function and classify the type of discontinuity it has at $x=3$.

Analyse the continuity of the function $f(x) = \frac{x}{1-e^{1/x}}$ at the point $x=0$. Calculate the one-sided limits and determine whether the discontinuity is removable or a jump discontinuity.

In a Spanish city, the on-street regulated parking service (blue zone) applies the following rate structure during the first two hours of parking:

• Up to 30 minutes (inclusive): €$0,60$.

• More than 30 and up to 60 minutes (inclusive): €$1,30$.

• More than 60 and up to 90 minutes (inclusive): €$2,10$.

• More than 90 and up to 120 minutes (inclusive): €$3,00$.

Let $P(t)$ be the function that describes the price to pay in euros according to the elapsed time $t$, measured in minutes, for $t \in (0, 120]$.

1. a) Write the analytical expression of the piecewise function $P(t)$.
2. b) Study analytically the continuity of $P(t)$ at the point $t = 60$: calculate $\lim_{t \to 60^-} P(t)$, $\lim_{t \to 60^+} P(t)$, and $P(60)$, and classify the discontinuity, indicating the jump size.
3. c) Interpret the practical meaning of this discontinuity for a driver who picks up their vehicle at exactly minute $60$ compared to another who arrives at minute $60,1$ ($60$ minutes and $6$ seconds).

Construct the analytical expression of a piecewise-defined function $f:\mathbb{R} \to \mathbb{R}$ that simultaneously satisfies the following four conditions:

1. It is continuous on all of $\mathbb{R}$ except at $x = -2$ and at $x = 3$.

2. At $x = -2$ it has a removable discontinuity, satisfying $\lim_{x \to -2} f(x) = 4$, but with $f(-2) = 0$.

3. At $x = 3$ it has a jump discontinuity (finite jump), such that the jump between the branches is exactly $5$ units; that is, $|\lim_{x \to 3^+} f(x) - \lim_{x \to 3^-} f(x)| = 5$.

4. The two branches that meet at $x = 3$ consist of straight lines (polynomial functions of degree 1).

Write the complete formula for $f(x)$ indicating the intervals of definition and formally prove, by calculating the one-sided limits and the value of the function at each critical point, that all the required conditions are met.

Determine the values of the parameters $a$ and $b$ so that the following function is continuous everywhere in $\mathbb{R}$:

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

Let $f$ be a continuous function on the closed interval $[0, 1]$ such that $0 \le f(x) \le 1$ for all $x \in [0, 1]$. Using Bolzano's Theorem, prove that there exists at least one point $c \in [0, 1]$ at which the graph of the function intersects the parabola $y = x^2$, that is, prove that $f(c) = c^2$. (Hint: Define an auxiliary function $g(x) = f(x) - x^2$).

To design a roller coaster, two sections of track must be joined. The profile of the roller coaster is given by the following piecewise function, where $x$ is the horizontal distance in meters:

$f(x) = \begin{cases} \frac{100}{x + 10} + a & \text{if } 0 \le x < 10 \\ bx^2 - 2x + 15 & \text{if } x \ge 10 \end{cases}$

Knowing that the design requires the track to be continuous at the joining point $x = 10$, and that Section 1 starts from a height of $15\text{ m}$ at $x = 0$, calculate the values of $a$ and $b$.

Discuss the continuity of the function $f(x) = \frac{x^2 - 1}{x^2 - mx + 4}$ depending on the possible values of the real parameter $m$. Pay special attention to the cases where the denominator has roots that coincide with those of the numerator.

Given the functions $g(x) = \frac{1}{x - 2}$ and $h(x) = \sqrt{x + 1}$, obtain the algebraic expression for the composite function $f(x) = g(h(x))$ and thoroughly study its continuity, stating its domain and the types of discontinuities it has.

The specific heat capacity $C_p(T)$ of a material undergoes a phase transition at a critical temperature $T_c = 100$ K. The proposed thermodynamic model is:

$C_p(T) = \begin{cases} \frac{500}{\sqrt{120-T}} & \text{if } T < 100 \\ 200 \ln(T-99) + k & \text{if } T \ge 100 \end{cases}$

Calculate the value of $k$ knowing that during the phase transition a finite jump discontinuity (latent heat) occurs such that $\lim_{T \to 100^+} C_p(T) - \lim_{T \to 100^-} C_p(T) = 80$.

Find the values of $a, b$ and $c$ so that the rational function $f(x) = \frac{ax^2+bx+c}{x^2-4}$ simultaneously satisfies the following conditions:

1. It has a removable discontinuity at $x=2$.

2. It has a vertical asymptote at $x=-2$.

3. Its limit as $x \to \infty$ is $3$.

4. It intersects the y-axis at $y = -1,5$.

Study the continuity of the function $f(x) = e^{\frac{1}{|x|-1}}$ in detail. Determine its domain, calculate the limits at the problem points, and rigorously classify the type of discontinuity present at each of them.

The voltage signal on an oscilloscope shows a damped wave that suddenly stabilises due to the action of a clipping diode at $t = t_c$.

The analytical model for $t \ge 0$ is:

$V(t) = \begin{cases} e^{-0,5t} \cos(2\pi t) & \text{if } 0 \le t < t_c \\ C & \text{if } t \ge t_c \end{cases}$

Given that the cutoff occurs exactly at the fourth complete cycle of the cosine wave ($t_c = 4$), prove analytically that the function is continuous at that point by calculating the exact value that the constant $C$ must take.

The goal is to determine rigorously whether the cubic equation $x^3 - 3x + 1 = 0$ has at least one real root located in the open interval $(0, 1)$. Two students propose different approaches:

• Method 1 (Marcos): Aims to solve the equation exactly by looking for integer or rational roots using Ruffini's rule among the divisors of the constant term ($\pm 1$), in order to then factor and solve for the unknown.

• Method 2 (Sara): Defines the auxiliary function $f(x) = x^3 - 3x + 1$ and proposes verifying the hypotheses of Bolzano's Theorem on the closed interval $[0, 1]$ to deduce the existence of the solution.

1. a) Carry out Marcos's attempt (Method 1). Does he manage to factor the polynomial using Ruffini's rule? Explain the limitation of elementary algebraic methods when dealing with this type of equations.
2. b) Carry out Sara's procedure (Method 2): state the hypotheses required by Bolzano's Theorem, rigorously check whether they hold for $f(x)$ on $[0, 1]$, and deduce the final conclusion.
3. c) Evaluate both methods: justify which one effectively achieves the proposed goal and reflect on the role that Bolzano's Theorem plays in mathematical analysis when exact algebraic solutions are not available.

During a winter thermal inversion episode recorded by AEMET in the northern peninsular plateau, the air temperature $T$ (in $^{\circ}\text{C}$) as a function of altitude $h$ (measured in hundreds of metres, with $h \ge 0$) is described by the following model:

where $a$ and $b$ are real parameters determined by atmospheric conditions.

1. a) For the model to correspond to a real physical phenomenon, the temperature cannot have abrupt discontinuities in the inversion layer ($h = 4$). Derive the algebraic equation that $a$ and $b$ must satisfy to ensure that $T(h)$ is continuous at $h = 4$.
2. b) Knowing that at ground level ($h = 0$) the temperature is $-3\ ^{\circ}\text{C}$ and that at an altitude of $900\text{ m}$ ($h = 9$) the thermometer reads $1,5\ ^{\circ}\text{C}$, calculate the numerical values of $a$ and $b$ so that the function is continuous throughout its domain $[0, +\infty)$.
3. c) With the values found, use Bolzano's Theorem to justify whether there is at least one altitude within the interval $[0, 4]$ at which the temperature is exactly $0\ ^{\circ}\text{C}$ (freezing level). Next, reason whether it would have been legitimate to apply Bolzano's Theorem directly on the overall interval $[0, 9]$ had the values of $a$ and $b$ not guaranteed continuity at $h = 4$.