Continuity of functions
When you walk up a hill, your height changes without any jumps: at every instant there is a height, and a moment later you are at almost the same one. But the price of hourly parking does jump: at $2{,}99$ hours you pay one rate and at $3{,}00$ hours, all of a sudden, a higher one. The first situation is continuous; the second is not. In this section you will turn that intuition into a precise criterion, using the limits you already know.
Continuity at a point
A function is continuous at $x=a$ if its graph does not break there: the value the function "approaches" from both sides is the same as the value it actually takes at $a$.
Express the function $h(x) = |x - 3|$ as a piecewise-defined function and prove analytically that it is continuous at $x = 3$.
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 logarithmic function $f(x) = \ln(8 - 2x)$, clearly stating the interval of real numbers where it is continuous.
Types of discontinuity
When any of the three conditions fails, the function is discontinuous at that point. We will study three common types. They do not cover every case: $\sin(1/x)$ oscillates without a limit as $x\to0$, and its discontinuity is not one of these three types.
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.
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…
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…
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?
Continuity of piecewise functions
If each expression is continuous on its own piece, you only need to check the joining points as well. If not, you must also study the interior discontinuities of each expression. There you have to compare the one-sided limits.
The charge for a public car park 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…
The charge for a public car park 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 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 car park user.
You can do it on paper and send a photo with your answer.
The temperature of an industrial oven in degrees Celsius, $T(t)$, as a function of time in minutes $t$, is given by:…
The temperature of an industrial oven in degrees Celsius, $T(t)$, as a function of time in minutes $t$, is given by:
$T(t) = \begin{cases} 20 + 15t & \text{if } 0 \le t < 10 \\ k \cdot 0,9^{t-10} & \text{if } t \ge 10 \end{cases}$
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?
You can do it on paper and send a photo with your answer.
Finding parameters so that a piecewise function is continuous
The most typical problem: given an unknown parameter in one of the expressions, find the value that removes the jump.
A mobile phone company offers a data tariff with the following conditions: a fixed fee of $10$ € is paid, which includes the first $5$ GB. For each additional GB up to…
A mobile phone company offers a data tariff 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 charged. Beyond $20$ GB, 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 algebraic expression for this function.
Determine the values of the parameters $a$ and $b$ so that the following function is continuous everywhere in $\mathbb{R}$:…
Determine the values of the parameters $a$ and $b$ so that the following function is continuous everywhere in $\mathbb{R}$:
$f(x) = \begin{cases} ax^2 + bx + 1 & \text{if } x \le 1 \\ 2a \ln(x) + b & \text{if } 1 < x \le e \\ e^{x-e} + a & \text{if } x > e \end{cases}$
Bolzano's Theorem
If a continuous function goes from negative to positive values (or the other way round) on an interval, it is because at some point in between it must have crossed zero. Continuity — having no jumps — guarantees that it cannot "pass over" the axis without touching it.
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…
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…
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…
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 $g(x) = f(x) - x^2$).