Exercises: Continuity of a function at a point

Observe the following graph of the function $f(x)$ and analyse its behaviour 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.

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.

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 metres) 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.

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

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

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 behaviour 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.

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.

In a 2nd year of Bachillerato exam, students are asked to study the continuity at the point $x = 1$ of the function:

Marcos writes the following answer in his exercise:

"Since substituting $x = 1$ into the first branch yields $\frac{0}{0}$, I factor the numerator as $(x - 1)(x + 1)$. I cancel the factor $(x - 1)$ with the denominator $|x - 1|$, so that the simplified expression is $x + 1$. Calculating the limit as $x \to 1$, I get $1 + 1 = 2$. Given that this limit coincides with $f(1) = 2$, the function is continuous at $x = 1$".

Your task is to:

1. a) Identify and explain the two conceptual errors made by Marcos (one algebraic regarding the cancellation of the absolute value, and another regarding the formal definition of limit).
2. b) Correct the exercise: write the piecewise function by removing the absolute value bars in a neighbourhood of $x = 1$, rigorously calculate the one-sided limits $\lim_{x \to 1^-} f(x)$ and $\lim_{x \to 1^+} f(x)$, and reach a reasoned conclusion on whether $f$ is continuous at $x = 1$, stating the type of discontinuity if it is not.

Consider the real function defined by:

1. a) Estimate: Before performing any formal algebraic development, approximate the behaviour of the difference quotient near the origin. To do this, numerically evaluate the expression $\frac{\sqrt{x + 4} - 2}{x}$ at values close to zero from both sides (for example, with $x = 0,1$, $x = 0,01$, $x = -0,1$, and $x = -0,01$). From these results, formulate a reasoned hypothesis: do you think the function will be continuous at $x = 0$?
2. b) Check: Analytically prove your conjecture by resolving the indeterminate form of $\lim_{x \to 0} f(x)$ by multiplying by the conjugate expression. In an orderly manner, check whether the three conditions required by the definition of continuity at $x = 0$ are satisfied.
3. c) Assess: Compare the exact algebraic result with the numerical approximations you calculated at the beginning, and reflect on the risk of confirming the continuity of a function based exclusively on a finite table of values.

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.

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.

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 $x = -1$.

Classify all the discontinuities of the function $g(x)$ and determine its vertical asymptotes by analysing the behaviour 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?

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.

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.

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.

An express courier company operating in Spain sets the price of its domestic shipments based on the weight $x$ of the parcel (in kilograms). The billed rate $P(x)$ (in euros) is given by the following piecewise function:

To comply with consumer protection legislation and prevent a tiny increase in weight from penalising the user with a sudden jump in the bill, the company requires that the price function has no discontinuities in its domain.

1. a) Determine the real values of the parameters $a$ and $b$ that make the function $P(x)$ continuous at the transition points $x = 5$ and $x = 9$.
2. b) Analytically justify the continuity at both points by verifying that the three conditions of continuity are satisfied once the parameters have been found.
3. c) Interpret the practical meaning of the value $P(5)$ and explain what would happen commercially if there were a jump discontinuity at $x = 5$ with $\lim_{x \to 5^+} P(x) = 15$ €. Who would be disadvantaged by this situation if the parcel weighs $5,01\text{ kg}$?

Design and write the algebraic expression of a piecewise function $f: \mathbb{R} \to \mathbb{R}$ that simultaneously satisfies the following three requirements:

1. It has a removable discontinuity at the point with abscissa $x = -2$.

2. It is continuous at the point with abscissa $x = 0$, having to change its algebraic expression at that point and mandatorily incorporating the absolute value $|x|$ into at least one of its branches.

3. It has a jump discontinuity with a jump of exactly 3 units at the point with abscissa $x = 3$ (that is, $|\lim_{x \to 3^+} f(x) - \lim_{x \to 3^-} f(x)| = 3$).

Once your function has been proposed:

1. a) Write its complete analytical formula, detailing the intervals of definition for each branch.
2. b) Justify step by step, applying the three conditions of the formal definition of continuity and through the calculation of the corresponding one-sided limits, why your function strictly satisfies each of the three imposed requirements.

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

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

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.

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.

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:

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

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.

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 $x = 1$.

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.

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?

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:

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

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

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.

We wish to study whether there is any value of the parameter $L \in \mathbb{R}$ for which the following function is continuous at the point $x = 1$:

To calculate the limit of the function at $x = 1$, two students propose two different approaches:

• Method 1 (Prior breakdown): Rewrite the function by removing the absolute value bars according to the sign of $(x - 1)$ to obtain a piecewise function in a neighbourhood of $x = 1$, simplify each branch algebraically, and evaluate the one-sided limits.

• Method 2 (Direct sign substitution): Keep the compact expression and directly calculate the one-sided limits by substituting $|x - 1|$ with $-(x - 1)$ when approaching from the left ($x \to 1^-$) and with $+(x - 1)$ when approaching from the right ($x \to 1^+$).

A third student suggests Method 3: Directly apply L'Hôpital's rule to the general quotient using the derivative of the absolute value $|u|' = \frac{u}{|u|} \cdot u'$.

1. a) Reasonably rule out Method 3 by explaining what geometric and analytical property fails at $x = 1$ for the function $|x - 1|$, which invalidates the use of the derivative at that point.
2. b) Apply both Method 1 and Method 2 to calculate the one-sided limits $\lim_{x \to 1^-} f(x)$ and $\lim_{x \to 1^+} f(x)$. Justify which of the two methods provides greater logical guarantees to avoid sign errors with the denominator.
3. c) From the results obtained, deduce whether it is possible to find the value of $L$ that makes the function continuous at $x = 1$. If it is not possible, classify the existing discontinuity, stating its jump.

Consider the function dependent on the real constants $a, b, c, k \in \mathbb{R}$:

1. a) Reasoning about finite limits: Note that the denominator of the first branch vanishes at $x = 2$. Reason why, for the function to have any possibility of being continuous at $x = 2$, it is strictly mandatory that the numerator also vanishes at $x = 2$. What type of behaviour would the function exhibit if that numerator were not zero?
2. b) Deduction of parameters by type of discontinuity: It is known that, in the interior of the interval $(-\infty, 2)$, the function has a removable discontinuity at the point $x = -2$. With this information and the deduction from the previous section, uniquely determine the values of the coefficients $a$ and $b$.
3. c) Welding of branches: With the values of $a$ and $b$ already calculated, find the value of the one-sided limit $\lim_{x \to 2^-} f(x)$. Next, calculate the right-hand limit $\lim_{x \to 2^+} f(x)$ using equivalent infinitesimals or L'Hôpital's rule. Finally, deduce what values $c$ and $k$ must take to ensure that $f$ is continuous at $x = 2$.