Continuity of Functions

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

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

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

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

Look at the following graphical representation of a function $f(x)$ and determine the points where it has discontinuities, classifying them as removable, finite jump, or infinite jump discontinuities.

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 so that the function is continuous at that point?

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.

Study the continuity of the trigonometric function $g(x) = \frac{1}{\cos x}$ in the open interval $(0, 2\pi)$. At which points does it have discontinuities and what types are they?

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

Given the function with modulus $f(x) = \frac{x^2-9}{|x-3|}$, rewrite it as a piecewise-defined function and classify the type of discontinuity it has at $x=3$.

Analyse the continuity of the function $f(x) = \frac{x}{1-e^{1/x}}$ at the point $x=0$. Calculate the one-sided limits and determine whether the discontinuity is removable or a jump discontinuity.

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

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

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

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

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?

You can do it on paper and send a photo with your answer.

A cell phone company offers a data plan with the following conditions: a fixed fee of $200$ pesos is paid, which includes the first $5$ GB. For each additional GB up to $20$ GB, $40$ pesos is charged. Beyond $20$ GB, the price follows the expression $a + 30x$, 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 algebraic expression for this function.

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?

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

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

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

The equation of the profile is:

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

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

The specific heat capacity $C_p(T)$ of a material undergoes a phase transition at a critical temperature $T_c = 100$ K. The proposed thermodynamic model is:

$C_p(T) = \begin{cases} \frac{500}{\sqrt{120-T}} & \text{if } T < 100 \\ 200 \ln(T-99) + k & \text{if } T \ge 100 \end{cases}$

Calculate the value of $k$ knowing that during the phase transition a finite jump discontinuity (latent heat) occurs such that $\lim_{T \to 100^+} C_p(T) - \lim_{T \to 100^-} C_p(T) = 80$.

The voltage signal on an oscilloscope shows a damped wave that suddenly stabilises due to the action of a clipping diode at $t = t_c$.

The analytical model for $t \ge 0$ is:

$V(t) = \begin{cases} e^{-0.5t} \cos(2\pi t) & \text{if } 0 \le t < t_c \\ C & \text{if } t \ge t_c \end{cases}$

Given that the cutoff occurs exactly at the fourth complete cycle of the cosine wave ($t_c = 4$), prove analytically that the function is continuous at that point by calculating the exact value that the constant $C$ must take.

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.

Given the functions $g(x) = \frac{1}{x - 2}$ and $h(x) = \sqrt{x + 1}$, obtain the algebraic expression for the composite function $f(x) = g(h(x))$ and thoroughly study its continuity, stating its domain and the types of discontinuities it has.

Find the values of $a, b$ and $c$ so that the rational function $f(x) = \frac{ax^2+bx+c}{x^2-4}$ simultaneously satisfies the following conditions:

1. It has a removable discontinuity at $x=2$.

2. It has a vertical asymptote at $x=-2$.

3. Its limit as $x \to \infty$ is $3$.

4. It intersects the y-axis at $y = -1.5$.

Study the continuity of the function $f(x) = e^{\frac{1}{|x|-1}}$ in detail. Determine its domain, calculate the limits at the problem points, and rigorously classify the type of discontinuity present at each of them.

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

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.

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