Continuous functions. Discontinuities

A piecewise function is given by the following expression:

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

Calculate the value of the function at $x=1$ and the one-sided limits as $x$ approaches 1. Is the function continuous at that point? If not, classify the discontinuity.

Observe the following graph of the function $f(x)$ and determine at which points of its domain it has a discontinuity. Classify each of them, stating whether it is a removable discontinuity, a finite jump discontinuity or an infinite discontinuity.

Observe the graph of the function $g(x)$ and answer the following questions:

1. Determine the domain of the function.

2. Identify the points of discontinuity and classify them.

3. Write the value of the one-sided limits at $x = -1$ and at $x = 3$.

Given the rational function $f(x) = \frac{3x-1}{x+5}$, state its domain of definition and the point at which it has a discontinuity. What type of discontinuity is it? Justify your answer by studying the behaviour of the function around that point.

Consider the function $f(x) = \frac{x^2-16}{x-4}$.

Calculate the value of the function at $x=4$ if possible. Next, simplify the algebraic expression and calculate the limit as $x$ approaches 4. What type of discontinuity does the function show at $x=4$?

Find the points of discontinuity of the function $f(x) = \frac{x+2}{x^2-3x-10}$ and state what type each one is, analysing whether or not the numerator is zero at those points.

Given the function $f(x) = \frac{x^2-x-2}{x^2-1}$, determine its domain of definition. Analytically study the type of discontinuity it presents at $x=1$ and at $x=-1$ by calculating the corresponding one-sided limits.

In a city bicycle car park, the charge is 2 € for the first hour or fraction thereof, and 1.50 € for each additional hour or fraction thereof. If we plot the function relating continuous parking time (in hours) to the price to pay (in euros), what type of discontinuity will the graph show at integer values of time ($t=1, 2, 3\dots$)?

Study the continuity at $x=0$ of the following function:

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

Classify the existing discontinuity at that point.

Calculate the value that parameter $k$ must have for the following function to be continuous throughout its domain:

$f(x) = \begin{cases} x^2 - 3x + k & \text{if } x \le 2 \\ 5x - 7 & \text{if } x > 2 \end{cases}$

An electricity company offers a tariff for SMEs: a fixed 15 € plus 0.12 € per kWh consumed, up to a maximum of 100 kWh. If consumption exceeds 100 kWh, the tariff changes and ALL consumed kWh are charged at 0.10 € (removing the fixed cost of 15 €).

Write the piecewise function determining the total cost as a function of the consumed kWh. Is the function continuous at an exact consumption of 100 kWh? If not, state the type of discontinuity and the "jump" in price.