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

Observe the graph of the function $f(x)$ represented below and identify the points of discontinuity. For each of them, indicate what type it is (removable, finite jump or infinite jump).

Calculate the value of the parameter $k$ so that the following piecewise function is continuous over 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 any discontinuities, classify them by calculating their one-sided limits and justifying your answer.

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?

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

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

1. Explain, using the Intermediate Value Theorem (IVT), whether the hiker has passed the 1,500-foot mark at any point.

2. Can we guarantee that they have passed the 2,000-foot mark?

Explain whether the function $f(x) = \frac{1}{x}$ necessarily reaches an absolute maximum and minimum in the following intervals based on the Extreme Value Theorem (EVT):

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

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

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

Indicate whether the function is continuous at $x=1$ by comparing its one-sided limits.

Consider the piecewise function $f(x)$ defined as:

Calculate the values of $a$, $b$ and $k$ so that the function is continuous throughout its domain $\mathbb{R}$. Once found, explain whether the function satisfies the hypotheses of Weierstrass's Theorem in the interval $[1, 5]$.

Study the continuity of the following function and classify all its points of discontinuity, justifying your answer:

Indicate if 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 feet 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 present? Interpret the result in the context of the animation (is there a jump, does the object disappear, or does it go to infinity?).
3. c) Is it possible for the object to reach a height of $0$ feet at any instant 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 at least one point within the interval $[0, \pi]$. Explain in detail the steps and hypotheses necessary to apply the theorem.

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

It is known that a function $f(x)$ is continuous in 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 meets 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 $k$ knowing that $\lim_{x \to 0} f(x) = 5$.
2. b) With 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 in the interval $[0, 2\pi]$. Can we guarantee that it reaches an absolute maximum value in this 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 $5$ of flight, the drone is exactly at a height of $2$ feet. Use the Intermediate Value Theorem (Darboux's Theorem) to do this.

Given the piecewise function $f(x)$ defined by the following expression, determine the values of the real parameters $a$ and $b$ so that the function is continuous over its entire domain. In the case that $b=1$, classify the type of discontinuity that the function presents 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 height (in feet) of drone $D_1$ with respect to the ground is given by the function $h_1(t) = t \cdot \ln(t+1)$, while the height of drone $D_2$ follows the function $h_2(t) = 10 \cdot e^{-t} + 5$, where $t$ is the time in minutes since take-off ($t \geq 0$).

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

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

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

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

Study the continuity of the following function in its domain of definition:

Determine the points where the function presents discontinuities and classify them. Investigate whether the function satisfies the hypotheses of Bolzano's Theorem in the interval $[0.5, 2]$ to ensure the existence of a root.

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

1. Express the function without moduli.

2. Study its continuity at the points where the arguments of the moduli are zero.

3. Using Weierstrass's Theorem, prove that the function reaches an absolute maximum in the closed interval $[-3, 3]$.

4. 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 is 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 angle 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 one-sided limits at the point where the function is not defined.

3. Classify the discontinuity found.

4. Is it possible to apply Bolzano's Theorem in the interval $[-1, 1]$ to prove that there is 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.