Continuity of a Function at a Point

Observe the following graph of the function $f(x)$ and analyse its behavior at the point with x-coordinate $x = 2$. State whether the function is continuous at that point by checking the three conditions for continuity: existence of $f(2)$, existence of $\lim_{x \to 2} f(x)$ and the equality of both values.

Check whether the function $f(x) = e^{x-1}$ is continuous at the point $x = 1$. For a complete answer, you must verify whether the value of the function at that point coincides with the limit of the function as $x$ approaches 1.

Study the continuity of the function $f(x) = \frac{x^2 - 1}{x - 1}$ at the point $x = 1$. If it has a discontinuity, state what type it is (removable, jump, or infinite) and explain whether there is any way to redefine the function so that it is continuous at that point.

Visually analyse the continuity of the function represented in the following graph. Identify the value of $x$ where the function has a discontinuity and classify it based on the behavior of its left-hand and right-hand branches.

Given the function $f(x) = \frac{1}{x + 4}$, justify why it is not continuous at the point $x = -4$. To do so, calculate the one-sided limits $\lim_{x \to -4^-} f(x)$ and $\lim_{x \to -4^+} f(x)$ and describe the type of discontinuity it has.

Given the following piecewise function, calculate the left-hand and right-hand limits at the boundary point $x = 3$ and determine whether the function is continuous at that point:

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

A surveillance drone follows an altitude trajectory defined by the function $h(t)$ (in meters) with respect to time $t$ (in seconds). Due to a change in the control system, the function changes at $t = 5$:

$h(t) = \begin{cases} t^2 - 10 & \text{if } 0 \le t < 5 \\ 3t & \text{if } t \ge 5 \end{cases}$

Check whether the transition in altitude is continuous at time $t = 5$ to prevent the drone from undergoing a sudden movement.

Study the continuity of the following function across its domain, classifying any points of discontinuity you find:

Accompany your analytical study with a diagram of the one-sided limits at the critical points.

Given the function:

Study its continuity at the points $x = 1$ and $x = -1$. If there are discontinuities, state what type they are and whether there is any way to redefine the function to remove them.

Study the continuity of the function $f(x) = \frac{1}{1 + 2^{1/x}}$ at the point $x = 0$. To do so, calculate the one-sided limits $\lim_{x \to 0^-} f(x)$ and $\lim_{x \to 0^+} f(x)$. What type of discontinuity does the function have at that point?

Study the continuity of the following function over its entire domain, classifying in detail each of its points of discontinuity:

$f(x) = \dfrac{|x^2 - 1|}{x^2 - 3x + 2}$

Afterwards, if there is any point where the discontinuity is removable, propose a redefinition of the function at that point to turn it into a function that is continuous there. If there is none, justify it.

Classify all the discontinuities of the function $g(x)$ and determine its vertical asymptotes by analysing the behavior of the function near the values where it is not defined:

How would you define the function at the points of removable discontinuity so that it becomes continuous at them?

Determine the value of the parameter $k$ so that the function $g(x)$ is continuous on the entire real line $\mathbb{R}$:

$g(x) = \begin{cases} x + k & \text{if } x \le 2 \\ x^2 - 1 & \text{if } x > 2 \end{cases}$

Given the function dependent on the parameter $k \in \mathbb{R}$:

Calculate the value of $k$ so that the function has a jump discontinuity (finite jump) at $x = 3$. Is there any value of $k$ that makes it continuous at that point? Justify your answer by computing the one-sided limits.

An industrial cooling system regulates its power $P$ (in kilowatts) according to the outside temperature $T$ (in degrees Celsius) by means of the following function:

Determine the value of the constant $a$ so that the power transition is continuous when the temperature passes through $0\,^\circ C$.

Consider the function $f(x)$ defined by:

Find the values of $a$ and $b$ so that the function is continuous on its domain. Once found, state whether the function is continuous at $x = -1$.

A new engine prototype is being tested whose acceleration $a$ (in $m/s^2$) depends on time $t$ (in seconds) according to the expression:

Calculate the value of $m$ so that the acceleration does not undergo sudden jumps (is continuous) at time $t = 2$ s.

Determine the values of $a$ and $b$ so that the function $f(x)$ is continuous everywhere in $\mathbb{R}$:

Explain in detail the process followed, checking that the left-hand and right-hand limits are equal at each boundary point.

A team of robotics engineers has programmed the motion of a mechanical arm. The speed of the end of the arm (in m/s) follows the path:

$v(t) = \begin{cases} \frac{\sqrt{t+5}-3}{t-4} & \text{if } 0 \leq t < 4 \\ \frac{a}{12} & \text{if } t = 4 \\ \frac{t^2 - 16}{t^2 - 3t - 4} \times b & \text{if } t > 4 \end{cases}$

Determine the values of $a$ and $b$ so that the robot's motion is continuous (without sudden jerks) at time $t = 4$ seconds.

A telecommunications company is adjusting the power of a 5G antenna. The signal loss $L$ (in decibels) depends on the distance $d$ (in km) according to the expression:

$L(d) = \begin{cases} \frac{k \times \ln(d+1)}{d} & \text{if } 0 < d < 1 \\ \sqrt{d^2 + 3} - a & \text{if } d \geq 1 \end{cases}$

The power transition is required to be smooth and continuous so as not to interrupt calls at the 1 km boundary. Calculate the values of $k$ and $a$ knowing that as the distance tends to 0 km, the signal loss tends to 2 dB.

In the design of a new component for a particle accelerator, the magnetic field strength as a function of position $x$ (in millimeters) is defined by the following function:

$f(x) = \begin{cases} \frac{e^{ax} - 1}{x} & \text{if } x < 0 \\ b & \text{if } x = 0 \\ \frac{x^2 - 4}{x - 2} \times \cos(\pi x) & \text{if } x > 0 \end{cases}$

Determine the values of the real parameters $a$ and $b$ so that the magnetic field strength is continuous at the point $x = 0$. Once found, would the function be continuous at the point $x = 2$? Reason your answer analytically.

Given the function $f(x) = \left( \frac{2x^2 + 3}{2x^2 - 1} \right)^{ax^2}$:

1. a) Find the value of the parameter $a \in \mathbb{R}$ such that $\lim_{x \to \infty} f(x) = e^2$.
2. b) With the value of $a$ obtained in the previous section, define a new function $g(x)$ such that:

$g(x) = \begin{cases} f(x) & \text{if } x \ge 1 \\ \frac{x^2 + bx + c}{x - 1} & \text{if } x < 1 \end{cases}$

Determine $b$ and $c$ so that $g(x)$ is continuous at $x = 1$.

Given the function depending on the parameter $k$:

$f(x) = \frac{x^3 - kx^2 + 3x}{x^2 - x}$

1. a) Find the value of $k$ so that the function has a removable discontinuity at $x = 1$.
2. b) For that value of $k$, study the continuity at $x = 0$ and determine whether any vertical asymptote exists.
3. c) Does any value of $k$ exist that makes the function continuous across all of $\mathbb{R}$? Justify your answer.

Apply Bolzano's Theorem to show that the graphs of the functions $f(x) = \ln(x)$ and $g(x) = e^{-x}$ intersect at least once within the interval $(1, e)$. Justify with reasoning whether this point of intersection is unique in that interval based on the growth of the functions.