Exercises: Continuity of functions

Analyze the continuity of the function $f(x) = \frac{x^2 - 16}{x - 4}$. State at which points it is discontinuous and classify the type of discontinuity it presents.

Given the piecewise function:

$f(x) = \begin{cases} 3x - 2 & \text{if } x \le 2 \\ x^2 & \text{if } x > 2 \end{cases}$

Determine analytically whether the function is continuous at the point $x = 2$.

Determine the domain of the function $g(x) = \sqrt{x^2 - 25}$ and justify in which intervals the function is continuous.

Observe the following graphical representation of a function $f(x)$ and determine the points where it presents discontinuities, classifying them into removable, finite jump or infinite jump.

Express the function $h(x) = |x - 3|$ as a piecewise function and prove analytically that it is continuous at $x = 3$.

Analyze the continuity of the rational function $f(x) = \frac{2x + 1}{x^2 + x - 6}$. Find the points where it is not continuous and specify the type of discontinuity in each case.

Given the exponential function $g(x) = e^{\frac{1}{x-1}}$, study its continuity at the point $x = 1$ by calculating the one-sided limits.

Check if the following function is continuous at $x = 0$:

$f(x) = \begin{cases} \frac{1}{x} & \text{if } x < 0 \\ x^2 + 1 & \text{if } x \ge 0 \end{cases}$

Find the value of the parameter $k$ so that the following function is continuous throughout its domain:

$f(x) = \begin{cases} \frac{x^2 - 9}{x + 3} & \text{if } x \ne -3 \\ k & \text{if } x = -3 \end{cases}$

The growth of a bacteria population in a laboratory culture is modeled by the function $P(t)$, where $t$ is the time in hours and $P(t)$ is the population in thousands of bacteria:

$P(t) = \begin{cases} 50(1 - e^{-0.2t}) & \text{if } 0 \le t < 10 \\ \frac{C}{t} & \text{if } t \ge 10 \end{cases}$

Calculate the value of the constant $C$ so that the population model is a continuous function at $t = 10$.

The fee for a public parking lot is calculated using the function $C(t) = 2 + 1.5 \cdot \text{ENT}(t)$, where $t$ is the time in hours and $\text{ENT}(t)$ represents the integer part function of $t$.

Identify the points of discontinuity of this function in the interval $(0, 4)$ and explain the real meaning of these discontinuities for a parking lot user.

Calculate the value of $a$ so that the function $f(x)$ is continuous at $x = 1$:

$f(x) = \begin{cases} ax^2 - 3 & \text{if } x \le 1 \\ e^{x-1} + 2a & \text{if } x > 1 \end{cases}$

Prove, using Bolzano's Theorem, that the equation $x^3 + 2x - 4 = 0$ has at least one real solution in the interval $(1, 2)$. Justify all the necessary conditions to apply the theorem.

The temperature of an industrial oven in degrees Fahrenheit, $T(t)$, as a function of time in minutes $t$, is given by:

$T(t) = \begin{cases} 20 + 15t & \text{if } 0 \le t < 10 \\ k \cdot 0.9^{t-10} & \text{if } t \ge 10 \end{cases}$

Determine the value of $k$ so that there is no sudden jump in temperature at 10 minutes. What physical phenomenon does the function represent from $t = 10$ onwards?

Find the value of $k$ so that the function is continuous at $x = 4$:

$f(x) = \begin{cases} \frac{\sqrt{x} - 2}{x - 4} & \text{if } x > 4 \\ kx + \frac{1}{4} & \text{if } x \le 4 \end{cases}$

The following graph illustrates the principle of Bolzano's Theorem for a continuous polynomial function $f(x) = x^3 - 4x - 2$. By observing the graph and evaluating the function, find an interval of length 1 (that is, $[n, n+1]$ with $n \in \mathbb{Z}$) where the existence of a real root can be guaranteed, justifying your answer.

Determine the values of the parameters $a$ and $b$ so that the following function is continuous throughout $\mathbb{R}$:

$f(x) = \begin{cases} ax^2 + bx + 1 & \text{if } x \le 1 \\ 2a \ln(x) + b & \text{if } 1 < x \le e \\ e^{x-e} + a & \text{if } x > e \end{cases}$

A technology company estimates that its revenue $I(x)$ and its costs $C(x)$, both in millions of dollars and as a function of the years $x$ of operation, are given by:

$I(x) = 10 \ln(x + 1)$

$C(x) = e^{x - 2}$

Using Bolzano's Theorem, prove that there is an exact moment between the second and fifth year of operation ($x \in [2, 5]$) when the company reaches its break-even point (revenue equals costs).

Calculate the values of $a$ and $b$ so that the rational function $f(x) = \frac{x^2 + ax + b}{x^2 - 4}$ has a removable discontinuity at $x = 2$ and, furthermore, the limit of the function at that point is equal to $3$.

Let $f$ be a continuous function on the closed interval $[0, 1]$ such that $0 \le f(x) \le 1$ for all $x \in [0, 1]$. Prove, using Bolzano's Theorem, that there is at least one point $c \in [0, 1]$ where the graph of the function intersects the parabola $y = x^2$, that is, prove that $f(c) = c^2$. (Hint: Define an auxiliary function $g(x) = f(x) - x^2$).

Find the values of $a$ and $b$ that make the following function, which includes an absolute value, continuous throughout its domain:

$f(x) = \begin{cases} \frac{|x - 2|}{x - 2} + ax & \text{if } x < 2 \\ b & \text{if } x = 2 \\ x^2 - ax + 1 & \text{if } x > 2 \end{cases}$

The battery charge level of an electric vehicle, $B(t)$ in percent, decreases continuously during a trip. At the initial instant $t = 0$ hours, the battery is at $100\%$. Upon arriving at the destination at $t = 5$ hours, the battery shows $20\%$. An engineer claims that, at some exact instant during the trip, the battery percentage coincided numerically with the expression $50 + 5t$.

Prove mathematically that the engineer is right using continuity theorems.

For the design of a roller coaster, two sections of track must be joined. The profile of the roller coaster is given by the following piecewise function, where $x$ is the horizontal distance in feet:

The equation of the profile is:

$f(x) = \begin{cases} \frac{100}{x + 10} + a & \text{if } 0 \le x < 10 \\ bx^2 - 2x + 15 & \text{if } x \ge 10 \end{cases}$

Knowing that the design requires the track to be continuous at the joining point $x = 10$, and that Section 1 starts from a height of $15\text{ ft}$ at $x = 0$, calculate the values of $a$ and $b$.

Study the continuity of the function $f(x)$ and determine the value of $k$ so that it is continuous at $x = 0$:

$f(x) = \begin{cases} \frac{1 - \sqrt{1 - x^2}}{x^2} & \text{if } x < 0 \\ k & \text{if } x = 0 \\ \frac{\sin(x^2)}{2x^2} & \text{if } x > 0 \end{cases}$