Continuity. Types
A cloud tariff that does not change price when it crosses a certain gigabyte threshold; another that does so all at once. The difference between the two situations is continuity, and now that you have mastered one-sided limits and the calculation of limits you can turn it into an exact criterion, with a precise classification of what happens when it fails.
Continuity at a point: the three conditions
A function is continuous at $x=a$ when there is no break in its graph right there: the value it approaches from both sides coincides with the value it actually takes.
Given the function $f(x) = \frac{x^2 - 9}{x - 3}$, state whether the function is defined at $x = 3$. Does the limit $\lim_{x \to 3} f(x)$ exist? Based on your answers,…
Given the function $f(x) = \frac{x^2 - 9}{x - 3}$, state whether the function is defined at $x = 3$. Does the limit $\lim_{x \to 3} f(x)$ exist? Based on your answers, decide whether the function is continuous at that point.
A home delivery app charges a fixed fee depending on the distance from the restaurant. The function defining the delivery cost in euros, $f(x)$, according to the…
A home delivery app charges a fixed fee depending on the distance from the restaurant. The function defining the delivery cost in euros, $f(x)$, according to the distance $x$ in kilometres is:
$$f(x) = \begin{cases} 2 & \text{if } 0 \leq x \leq 3 \\ 4 & \text{if } x > 3 \end{cases}$$Check whether the function is continuous at the point $x = 3$ by calculating its one-sided limits.
Classification of discontinuities
When any of the three conditions fails, there are three different ways in which it can fail, and each has its own name and graphical appearance.
Classify the type of discontinuity presented by the function shown in the image at the point $x = 2$. Justify whether it is a removable discontinuity, a finite jump…
Classify the type of discontinuity presented by the function shown in the image at the point $x = 2$. Justify whether it is a removable discontinuity, a finite jump discontinuity, or an infinite jump discontinuity.
Study the continuity of the following rational function across its domain and classify its points of discontinuity: $$f(x) = \frac{x^2 - x - 6}{x^2 - 9}$$ For the points…
Study the continuity of the following rational function across its domain and classify its points of discontinuity:
$$f(x) = \frac{x^2 - x - 6}{x^2 - 9}$$For the points where the discontinuity is removable, define a new function $g(x)$ that is continuous at that point.
Continuity in piecewise functions: finding parameters
The most common type of exercise in the university entrance exam (EBAU): given one or two parameters in a piecewise function, find their values so that it is continuous on all of $\mathbb{R}$.
Determine the values of the parameters $a$ and $b$ so that the following piecewise function is continuous over the whole real line $\mathbb{R}$: $$f(x) = \begin{cases} x…
Determine the values of the parameters $a$ and $b$ so that the following piecewise function is continuous over the whole real line $\mathbb{R}$:
$$f(x) = \begin{cases} x + 2a & \text{if } x < -1 \\ 3x + b & \text{if } -1 \leq x < 2 \\ x^2 - 5 & \text{if } x \geq 2 \end{cases}$$Given the function depending on the real parameters $k$ and $m$: $$g(x) = \begin{cases} \frac{k \cdot \text{sen}(x)}{x} & \text{if } x < 0 \\ x^2 + m & \text{if } 0 \le…
Given the function depending on the real parameters $k$ and $m$:
$$g(x) = \begin{cases} \frac{k \cdot \text{sen}(x)}{x} & \text{if } x < 0 \\ x^2 + m & \text{if } 0 \le x \le 1 \\ \frac{\sqrt{x} - 1}{x - 1} & \text{if } x > 1 \end{cases}$$Find the values of $k$ and $m$ so that $g(x)$ is a continuous function on the set of all real numbers.
Continuity of rational functions on their whole domain
In a rational function, the only candidates for discontinuity are the points that make the denominator vanish. Factorising numerator and denominator decides, at each one, whether the discontinuity is removable or a vertical asymptote.
An express delivery company sets its monthly rates for corporate clients according to the volume of packages sent ($x$, in hundreds of units). The total cost function…
An express delivery company sets its monthly rates for corporate clients according to the volume of packages sent ($x$, in hundreds of units). The total cost function ($C(x)$, in thousands of euros) is given by:
$$C(x) = \begin{cases} \dfrac{x^2 + 2x}{x} & \text{if } 0 < x < 2 \\ k \cdot (x - 1) & \text{if } x \geq 2 \end{cases}$$Calculate the value of the parameter $k$ so that the function is continuous at $x = 2$. What type of discontinuity does the function present at the point $x = 0$?
Back to the start: The tariff that jumps in price when a data volume is exceeded has a finite jump discontinuity at that threshold, whereas the tariff with a progressive cost and no jumps models a continuous function.