Continuity

Section 3.5 Continuity

We encountered some functions whose values become arbitrarily large as

$x \to a\text{.}$ Those are examples of functions that are not “continuous”. Roughly speaking, a function is continuous if you can draw the graph of the function without lifting your pen. In this section we study the notion of continuity in more detail.

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Objectives

You should be able to:

Explain and illustrate the definition of continuity.
Explain and illustrate infinite, jump, and removable discontinuities.
Apply the definition of continuity to determine whether or not a function is continuous at a point.

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Subsection 3.5.1 Instructional video

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There is no instructional video on this particular topic. It will be covered during the live lecture.

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Subsection 3.5.2 Key concepts

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Concept 3.5.1. Continuity.

A function $f(x)$ is continuous at $x=a$ if

$\begin{equation*} \lim_{x \to a} f(x) = f(a). \end{equation*}$

Thus, for a function to be continuous, it must satisfy the three conditions:

$f(a)$ exists;
$\displaystyle \lim_{x \to a} f(x)$ exists;
$\displaystyle \lim_{x \to a} f(x) = f(a)\text{.}$

A function is continuous on an interval if it is continuous at every point in the interval. Practically, it is continuous over an interval if you can draw the graph of the function over this interval without lifting your pen.

A function is continuous from the right at $x=a$ if

$\begin{equation*} \lim_{x \to a^+} f(x) = f(a), \end{equation*}$

while it is continuous from the left at $x=a$ if

$\begin{equation*} \lim_{x \to a^-} f(x) = f(a), \end{equation*}$

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Concept 3.5.2. Combining continuous functions.

Let $a, c \in \mathbb{R}\text{,}$ and $f(x), g(x)$ be functions that are continuous at $x=a\text{.}$ Then the following functions are also continuous at $x=a\text{:}$

$\begin{align*} \amp f(x) + g(x),\\ \amp f(x)-g(x),\\ \amp c f(x),\\ \amp f(x) g(x),\\ \amp \frac{f(x)}{g(x)} \qquad \text{provided }g(a) \neq 0 . \end{align*}$

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Concept 3.5.3. Some continuous functions.

The following functions are continuous at every point in their domain: polynomials, rational functions, root functions, trigonometric functions, exponential functions, and logarithmic functions. Most functions that appear in physics are continuous.

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Concept 3.5.4. Composition of continuity.

Let $f(x)$ be a function that is continuous at $x=b\text{,}$ and $\displaystyle \lim_{x \to a} g(x) = b\text{.}$ Then

$\begin{equation*} \lim_{x \to a} f(g(x)) = f(b) = f \left( \lim_{x \to a} g(x) \right). \end{equation*}$

Thus, if $g(x)$ is continous at $x=a\text{,}$ we have that $\displaystyle \lim_{x \to a} g(x) = g(a).$ Therefore

$\begin{equation*} \lim_{x \to a} f(g(x)) = f(g(a)), \end{equation*}$

that is, the composition

$\begin{equation*} (f \circ g)(x) = f ( g (x) ) \end{equation*}$

is continuous at $x=a\text{.}$

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MATH 144: Calculus for the Physical Sciences I