Exercises: Continuity of a function on an interval. Classification of discontinuities. Bolzano's and Weierstrass's theorems

Observe the graph of the function $f(x)$ shown below and identify the points of discontinuity. For each one, state what type it is (removable, jump, or infinite).

Calculate the value of the parameter $k$ so that the following piecewise function is continuous on the set of all real numbers:

Given the rational function $f(x) = \frac{x-3}{x^2-9}$, study its continuity at the points $x = 3$ and $x = -3$. If there are discontinuities, classify them, giving reasons by calculating their one-sided limits.

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

1. Prove that the function is not continuous at $x = 2$.

2. What value should $f(2)$ take so that the discontinuity is removable and the function becomes continuous at that point?

Check whether the function $f(x) = x^3 - 3x + 1$ satisfies the hypotheses of Bolzano's Theorem on the interval $[0, 1]$. Can you guarantee that there is at least one root of the equation $x^3 - 3x + 1 = 0$ in that interval? Justify your answer.

A hiker starts a route at a shelter at an altitude of 1 200 metres ($t=0$). After 4 hours of walking, they are at a viewpoint at an altitude of 1 800 metres. Assuming that the function describing their altitude with respect to time, $A(t)$, is continuous:

1. Explain, using the Intermediate Value Theorem, whether the hiker passed through the altitude of 1 500 metres at any point.

2. Can we guarantee that they passed through 2 000 metres?

Explain whether the function $f(x) = \frac{1}{x}$ must reach an absolute maximum and minimum on the following intervals, based on Weierstrass's Theorem:

1. On the closed interval $[1, 5]$.

2. On the open interval $(0, 1)$.

Study the one-sided continuity of the following function at the endpoints of the interval $[0, 4]$:

State whether the function is continuous at $x=1$ by comparing its left-hand and right-hand limits.

To study whether the function $f(x) = \frac{x+1}{x-2}$ vanishes in the interval $[1, 4]$, Alejandro writes the following in his exam:

"We compute the endpoints of the interval: $f(1) = \frac{1+1}{1-2} = -2 < 0$ and $f(4) = \frac{4+1}{4-2} = \frac{5}{2} > 0$. Since they have opposite signs, by Bolzano's Theorem there exists at least one value $c \in (1, 4)$ such that $f(c) = 0$. Furthermore, at $x = 2$ the function has a removable discontinuity because it only fails at an isolated point of the denominator".

1. a) Explain in a reasoned manner what serious mistake Alejandro makes when applying Bolzano's Theorem and analytically check whether the equation $f(x) = 0$ actually has any solution within the interval $(1, 4)$.
2. b) Rigorously classify the discontinuity of $f(x)$ at $x = 2$ by calculating the relevant one-sided limits and indicate which violated hypothesis prevented applying the theorem.

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

1) It has a removable discontinuity at $x = -2$, with $\lim_{x \to -2} g(x) = 3$.

2) It has a jump discontinuity (finite jump) at $x = 1$, such that the jump length between its one-sided limits is exactly 4 units.

3) It is continuous at every other point in $\mathbb{R}$.

Analytically justify that the function you designed satisfies each of the conditions by computing the corresponding one-sided limits and the function values at those points.

Let $f(x)$ be the piecewise function defined as:

The figure shows the three defining pieces (they change at $x=2$ and at $x=4$) and the interval $[1, 5]$.

1. a) Calculate the values of $a$, $b$ and $k$ so that the function is continuous on its entire domain $\mathbb{R}$.

1. b) Once you have found them, explain whether the function satisfies the hypotheses of the Weierstrass Theorem on the interval $[1, 5]$.

Study the continuity of the following function and classify all its points of discontinuity, giving reasons:

State whether any of the discontinuities is removable and, if so, redefine the function at that point so that it becomes continuous.

A graphic designer is plotting the trajectory of an object in an animation. The height $h$ in metres with respect to time $t$ (in seconds) is given by:

1. a) State the domain of the trajectory.
2. b) Study the continuity at $t = 3$. What type of discontinuity does it have? Interpret the result in the context of the animation (is there a jump, does the object disappear, or does it tend to infinity?).
3. c) Is it possible for the object to reach a height of $0$ metres at some time in the interval $[0, 2]$? Justify your answer using Bolzano's Theorem.

Prove, using Bolzano's Theorem, that the graphs of the functions $f(x) = e^x - 2$ and $g(x) = \cos(x)$ intersect at least at one point in the interval $[0, \pi]$. Explain in detail the steps and hypotheses required to apply the theorem.

Given the function $f(x) = x^3 + x - 5$, justify whether there exists a value $c \in [1, 2]$ such that $f(c) = -1$. What theorem guarantees this result? State the theorem and verify that its hypotheses are satisfied.

It is known that a function $f(x)$ is continuous on the closed interval $[0, 4]$. If $f(0) = 1$ and $f(4) = 1$, can it be guaranteed that there is a point $c \in (0, 4)$ such that $f(c) = 0$? Give a graphical example of a function that satisfies the conditions but has no roots in that interval, and explain why it does not contradict Bolzano's Theorem.

Given the function $f(x) = \frac{k \cdot \sin(x)}{x}$ for $x \neq 0$:

1. a) Find the value of $k$ given that $\lim_{x \to 0} f(x) = 5$.
2. b) Using that value of $k$, define $f(0)$ so that the function is continuous at $x = 0$.
3. c) Justify whether the resulting function is continuous on the interval $[0, 2\pi]$. Can we guarantee that it reaches an absolute maximum value in that interval?

A drone takes off from the ground ($y=0$) and follows a trajectory given by the height function $h(t) = t^2 \cdot e^{-t} + \ln(t+1)$, where $t$ is the time in minutes.

Prove that at some instant between minute $1$ and minute $6$ of flight, the drone is at an altitude of exactly $2$ meters. Use the Intermediate Value Theorem (Darboux's Theorem) for this purpose.

The temperature $T$ (in degrees Celsius) recorded by an AEMET weather station in a mountain area during a 12-hour winter day is modeled by the function:

where $t$ represents the elapsed time in hours since $08:00$ in the morning ($t = 0$).

1. a) Without the need to calculate derivatives, justify in a reasoned manner whether it can be guaranteed that at some point between $08:00$ and $20:00$ ($t = 12$) the thermometers marked exactly $0^\circ\text{C}$. State the theorem on which you rely and check in detail the fulfillment of all its hypotheses.
2. b) Do continuity theorems allow guaranteeing that throughout those 12 hours there was at least one instant of absolute maximum temperature and another of absolute minimum temperature? Cite the corresponding theorem, check its hypotheses, and interpret its practical meaning.

Consider the polynomial function $f(x) = x^3 + 5x - 3$ defined on the closed interval $[0, 1]$.

1. a) Estimate: Looking at the values of $f(0)$ and $f(1)$, as well as the relative weight of the linear term $5x$ compared to the cubic term $x^3$ on the interval $[0, 1]$, intuitively reason in which of the two subintervals, $[0;\, 0,5]$ or $[0,5;\, 1]$, the root of the function is more likely to be found.
2. b) Check: Rigorously apply Bolzano's Theorem to locate with certainty the root in one of the two previous subintervals. To do this, state its hypotheses, evaluate $f$ at the midpoint $x = 0,5$, and conclude with justification.
3. c) Assess whether your initial estimate matched the result obtained and reason whether it is possible for the function to have more than one solution in the overall interval $[0, 1]$ considering the monotonicity of $f(x)$.

Given the piecewise function $f(x)$ defined by the following expression:

1. a) Determine the values of the real parameters $a$ and $b$ so that the function is continuous on its entire domain.

1. b) In the case where $b=1$, classify the type of discontinuity that the function has at $x=0$ depending on the value of $a$.

A team of engineers is testing two new models of surveillance drones, $D_1$ and $D_2$. The altitude (in metres) of drone $D_1$ above the ground is given by the function $h_1(t) = t \cdot \ln(t+1)$, while the altitude of drone $D_2$ follows the function $h_2(t) = 10 \cdot e^{-t} + 5$, where $t$ is the time in minutes since takeoff ($t \geq 0$).

Use Bolzano's Theorem to prove that there is a time $t_0 \in [2, 5]$ at which both drones are at exactly the same altitude. Justify in detail whether you could guarantee, using Weierstrass's Theorem, that the difference in altitude between both drones reaches a maximum value in that time interval.

Consider the family of functions depending on the parameter $k \in \mathbb{R}$:

Analyse and classify all discontinuities of $f(x)$ depending on the possible values of $k$. State specifically for which values of $k$ any of the discontinuities is removable and redefine the function at that point so that it is continuous.

Reasonably prove that the equation $\sqrt{x^4 + 1} = 3x^2 \cdot \cos(x)$ has at least two real solutions in the interval $[-\pi, \pi]$. To do this, define an appropriate auxiliary function and apply the necessary continuity theorems. Is it possible to state that an absolute maximum value exists for the function $g(x) = \frac{1}{\sqrt{x^4+1} - 3x^2\cos(x)}$ in the interval $[0, \pi/4]$? Justify your answer based on the Weierstrass Theorem.

Study the continuity of the following function on its domain:

Determine the points where the function has discontinuities and classify them. Investigate whether the function satisfies the hypotheses of Bolzano's Theorem in the interval $[0,5; 2]$ to guarantee the existence of a root.

Let the function be $f(x) = |x^2 - 4| \cdot e^{-|x|}$. The figure shows its graph as a visual aid.

1. a) Express the function without absolute values, that is, defined piecewise.

1. b) Study its continuity at the points where the arguments of the absolute values vanish.

1. c) Using the Weierstrass Theorem, prove that the function attains an absolute maximum on the closed interval $[-3, 3]$.

1. d) Does the function have any horizontal asymptote? Relate this fact to the long-term behavior of the function.

Consider a continuous function $f:[0, 1] \to [0, 1]$. That is, the domain is the interval $[0, 1]$ and all its images are also contained in $[0, 1]$.

Prove, using Bolzano's Theorem, that there exists at least one point $c \in [0, 1]$ such that $f(c) = c$ (a fixed point of the function). Hint: Consider the auxiliary function $g(x) = f(x) - x$. Explain graphically what this result means in terms of the intersection of the graph of $f(x)$ with the line $y=x$ (the bisector of the first quadrant).

Given the function $f(x) = \frac{1}{1 + 2^{1/x}}$:

1. Find the domain of the function.

2. Calculate the left-hand and right-hand limits at the point where the function is not defined.

3. Classify the discontinuity found.

4. Is it possible to apply Bolzano's Theorem on the interval $[-1, 1]$ to prove that there exists a value where $f(x) = 0,5$? Explain whether the lack of continuity at a point invalidates the application of the theorem even if the values at the endpoints suggest that such a point should exist.

It is desired to prove that the equation $\cos(x) = x^2 - 2$ has at least two distinct real solutions in the closed interval $[-\pi, \pi]$. To address the proof, two strategies are proposed:

• Method A: Define the auxiliary function $h(x) = x^2 - 2 - \cos(x)$, analyze its symmetry (parity), and apply Bolzano's Theorem on suitable subintervals of $[0, \pi]$ to deduce the existence of both solutions.

• Method B: Attempt to isolate algebraically the variable $x$ through direct trigonometric transformations and the inverse function $\arccos(x)$ to compute the exact values of the intersection points.

1. a) Reasonably compare both methods. Justify why Method B is practically unfeasible for this transcendental equation and explain the advantages of Method A.
2. b) Fully and rigorously develop Method A: verify the hypotheses of Bolzano's Theorem, select a suitable subinterval within $[0, \pi]$ where the sign change is observed, and conclude why there must necessarily be at least two symmetric roots in $[-\pi, \pi]$.

To delve deeper into the necessity of the hypotheses of the fundamental theorems of continuity, solve the following analytical construction questions:

1. a) Design the analytical expression of a function $f:[0, 4] \to \mathbb{R}$ that simultaneously satisfies: $f(0) = -2$, $f(4) = 3$, $f(x) \neq 0$ for all $x \in [0, 4]$ (that is, it has no roots in the interval), and has a single discontinuity at $x = 2$. Classify said discontinuity and explain why the absence of roots does not contradict Bolzano's Theorem.
2. b) Is it possible to define a function $g:[0, 4] \to \mathbb{R}$ on that same closed and bounded interval that achieves NEITHER an absolute maximum NOR an absolute minimum? If you believe it is impossible, prove why; if you believe it is possible, construct an explicit analytical example of such a function $g(x)$, compute its image set (range), and explain which hypothesis of Weierstrass's Theorem has been violated.

Unit challenge. Where does uncertainty converge? Investigate the law of large numbers and process saturation by modelling a real phenomenon.

1. Data collection and sequence: In GeoGebra or a spreadsheet, carry out a simulation of $n=1000$ repetitions of a random event with $p=0,5$ (or download rainfall day frequencies from the AEMET website). Generate the sequence of relative frequencies $f_n$ and experimentally verify how $\lim_{n \to \infty} f_n = p$, connecting sequences with the frequentist interpretation of probability.

2. Model and asymptotes: Design a piecewise continuous function $P(t)$ that models this stabilisation over time. Formally study its continuity at the boundaries between pieces and determine its horizontal and vertical asymptotes, drawing conclusions about its long-term behaviour ($t \to +\infty$).

3. Indeterminate forms and L'Hôpital: Formulate the rate of convergence by calculating the derivative or the rate of change using a limit with a $\left[\frac{0}{0}\right]$ indeterminate form. Solve it by rigorously applying L'Hôpital's rule to obtain information about the growth of the function.

Submit a brief one-page report with the link to your digital file and the algebraic workings.

Assessment criteria:

• Accuracy in calculating limits and indeterminate forms using L'Hôpital's rule.

• Analytical justification of continuity at a point and deduction of asymptotes.

• Conceptual connection between the limit of the frequency sequence and theoretical probability.

• Rigour of the report and appropriate use of GeoGebra or the spreadsheet.