Continuity of a Function

Given the piecewise function:

$f(x) = \begin{cases} 2x^2 - 3 & \text{if } x < 2 \\ 5x - 5 & \text{if } x \ge 2 \end{cases}$

Determine if $f(x)$ is continuous at the point $x = 2$.

Observe the graph of the function $f(x)$. Identify the points where the function presents discontinuities and classify the type of discontinuity at each of them.

The rational function $f(x) = \frac{x^2 + 2x - 8}{x - 2}$ is not defined at $x = 2$. What numerical value should $f(2)$ take for the function to be continuous at this point?

Determine the domain of continuity of the function with radical $f(x) = \sqrt{x^2 - 5x + 6}$.

Study the continuity of the logarithmic function $f(x) = \ln(8 - 2x)$, clearly indicating the interval of real numbers where it is continuous.

Given the functions $g(x) = \frac{1}{x - 2}$ and $h(x) = \sqrt{x + 1}$, find the analytical expression of the composite function $f(x) = g(h(x))$ and exhaustively study its continuity, indicating its domain and the type of discontinuities it presents.

Study the continuity of the function $f(x) = \frac{x^2-x-6}{x^2-9}$ and classify its points of discontinuity.

Study in detail the continuity of the rational function $f(x) = \frac{x^3 - 3x^2 - x + 3}{x^2 - x - 6}$. Classify the types of discontinuity it presents at the points where it is not defined.

Calculate the value of $k$ so that the following function is continuous at $x = 1$:

$f(x) = \begin{cases} 3x + k & \text{if } x \le 1 \\ x^2 - 4x + 5 & \text{if } x > 1 \end{cases}$

Study the continuity of the following function, calculating the value of $k$ that makes it continuous at the point $x=0$:

$f(x) = \begin{cases} e^{2x} + k & \text{if } x \le 0 \\ \frac{x^2 + 3x}{x} & \text{if } x > 0 \end{cases}$

Determine the values of the parameters $a$ and $b$ so that the function $f(x)$ is continuous over the entire set of real numbers $\mathbb{R}$:

$f(x) = \begin{cases} x + a & \text{if } x \le -1 \\ x^2 - b & \text{if } -1 < x < 2 \\ 3x - 1 & \text{if } x \ge 2 \end{cases}$

Calculate the parameters $a$ and $b$ so that the following function is continuous over its entire domain:

$f(x) = \begin{cases} \frac{a}{x - 1} & \text{if } x < 0 \\ x^2 + b & \text{if } 0 \le x \le 2 \\ e^{x - 2} + 1 & \text{if } x > 2 \end{cases}$

The specific heat capacity of a gas as a function of temperature $T$ (in Kelvin) is modeled by the following piecewise function:

$c_p(T) = \begin{cases} a + bT & \text{if } 0 < T \le 100 \\ c \log_{10}(T) & \text{if } T > 100 \end{cases}$

Knowing that $c_p(10) = 3.5$, that the physical model requires the function to be continuous at $T = 100$, and that at a temperature of $1000$ Kelvin the capacity is $12$, determine the values of the parameters $a, b$ and $c$.

Given the function $f(x) = \frac{x^2 + kx - 6}{x - 2}$, calculate the value of the parameter $k$ so that the discontinuity at $x = 2$ is removable. Once found, what will be the value of the limit of the function at that point?

Find the values of the parameters $a, b, c$ and $d$ so that the rational function $f(x) = \frac{x^2 + ax + b}{x^2 - cx + d}$ simultaneously satisfies the following three conditions:

• It has a removable discontinuity at $x = 1$.

• It has an infinite jump discontinuity at $x = -2$.

• Its graph passes through the origin.

Discuss the continuity of the function $f(x) = \frac{x^2 - 1}{x^2 - mx + 4}$ depending on the possible values of the real parameter $m$. Pay special attention to the cases where the denominator has roots that coincide with those of the numerator.

Express the absolute value function $f(x) = |x^2 - 4x|$ as a piecewise function and study its continuity at the points $x = 0$ and $x = 4$.

A central parking garage charges $\$2.50$ for the first hour or fraction thereof, and $\$1.50$ for each additional hour or fraction thereof.

Write the analytical expression of the step function $C(t)$ representing the cost for a parking time of $0 < t \le 4$ hours. Study its continuity and state what type of discontinuities it presents.

Study the continuity of the function $f(x) = \frac{|x^2 - 4|}{x - 2}$. First write the function as a piecewise function without using the absolute value and classify in detail the discontinuities you find.

A cell phone company offers a data plan with the following conditions: a fixed fee of $\$10$ is paid which includes the first $5$ GB. For each additional GB up to $20$ GB, $\$2$ is paid. From $20$ GB onwards, the price follows the expression $a + 1.5x$, where $x$ is the total GB consumed.

Find the value of the parameter $a$ so that the function representing the total cost is continuous at all times and write the complete analytical expression of this function.

The following graph shows the concentration $C(t)$ of a drug in a patient's blood as a function of time $t$ (in hours). The curve for $0 \le t \le 4$ follows a quadratic model of the form $C(t) = at^2+bt$, and for $t > 4$ it follows an exponential model $C(t) = k \cdot 2^{-0.5t}$.

1. a) Determine the values of the parameters $a$, $b$ and $k$ by observing the key points on the graph.
2. b) Study the continuity of the function at $t=4$ and interpret the meaning of the discontinuity in the medical context of the drug's administration.

The graph shows the representation of the function $g(x) = \frac{x^2 + bx + c}{x^2 + dx + e}$ for all $x \neq -1$, additionally defining $g(-1) = 2$.

1. a) From the asymptotes and the hole observable in the graph, determine the values of the parameters $b, c, d$ and $e$.
2. b) Study the continuity of $g(x)$ over its entire domain and classify its discontinuities.

Use Bolzano's theorem to prove that the equation $x^3 - 3x + 1 = 0$ has at least one real solution in the interval $[0, 1]$. Explain why the theorem is applicable to this particular case.

Determine for which values of the parameter $m$ it can be guaranteed, by rigorously applying Bolzano's theorem, that the function $f(x) = x^3 - mx^2 + 2$ has at least one root in the closed interval $[-1, 1]$.