Continuity on an Interval. Discontinuities. The Bolzano and Weierstrass Theorems

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.

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

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 $[1, 5]$.

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).

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?

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.

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.

A graphic designer is plotting the trajectory of an object in an animation. The height $h$ in meters 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 $0$ meters at some time in the interval $[0, 2]$? Justify your answer using Bolzano's Theorem.

A hiker starts a route at a shelter at an altitude of 1.200 meters ($t=0$). After 4 hours of walking, they are at a viewpoint at an altitude of 1.800 meters. 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 meters at any point.

2. Can we guarantee that they passed through 2.000 meters?

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.

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.

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.

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

Prove that at some time between minute $1$ and minute $5$ of flight, the drone is at an altitude of exactly $2$ meters. Use the Intermediate Value Theorem to do so.

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

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 $(0, 1)$.

Given the function $f(x) = \frac{k \times \sin(x)}{x}$ 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?

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.

Let the function be $f(x) = |x^2 - 4| \times e^{-|x|}$. 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).

Reasonably prove that the equation $\sqrt{x^4 + 1} = 3x^2 \times \cos(x)$ 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.

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.

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 $a$.

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.