Calculus Notes

Section 1.4: Continuity

Continuity

In other words, the value that a function approaches as it nears a point is always equal to the value of the function at that point. A continuous function can be plotted without lifting one's pencil from the paper; there are no sudden jumps or holes.

A function is said to be continuous over the closed interval $[a,b]$ if it continuous over the open interval $(a,b)$ and in addition: $$\lim_{x \to a+} f(x) = f(a) \hspace{15pt} \text{and} \hspace{15pt} \lim_{x \to b-} f(x) = f(b)$$

In particular, all polynomial functions $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0$ are continuous for all real numbers. Functions of the form $f(x) = x^{m/n}$ are continuous on their domain---except possibly when $x = 0$.

If $f(x)$ is continuous at $c$ and $g(x)$ is continuous at $f(c)$, then the composite function $g\circ f = g(f(x))$ is continuous at $c$.

Discontinuities

• Removable discontinuity: The limit of a function exists (and is finite) at at , but the function does not exist at that point or has a value different from the limit. This type of discontinuity is called "removable", because changing the function could be made continuous by changing its value at that one point. Example: $$f(x) = \frac{x^2-1}{x-1} \hspace{10pt}\text{at}\hspace{10pt} x=1$$.
• Jump discontinuity: the left-hand and right-hand limits exist, but they do not equal each other. Example: $$f(x) = \begin{cases} 3 & x \ge 1 \\ -5 & x < 0 \end{cases} \hspace{10pt}\text{at}\hspace{10pt} x=1$$
• Infinite discontinuity: The function has a vertical asymptote (see next section). Example: $$ f(x) = \frac{1}{x} \hspace{10pt} \text{at}\hspace{10pt} x = 0$$
• Oscillating discontinuity: As $x$ approaches a particular number, the frequency of a sinusoidal function becomes infinite. Example: $$ \sin\left(\frac{1}{x}\right) \hspace{10pt} \text{at}\hspace{10pt} x = 0$$

Intermediate value theorem

The intermediate value theorem (IVM) states that "if $f(x)$ is continuous over the closed interval $[a,b]$, and $y$ is any number between $f(a)$ and $f(b)$, then there must exist some $x$-value $c$ such that $f(c) = y$."

In other words, if $y$ is a number between between $f(a)$ and $f(b)$, then the function must pass through $y$ on its way from $f(a)$ to $f(b)$ or else jump (have a discontinuity) along the way.

The IVM is often used to prove the exitence of roots. If a function is continuous and $f(a) < 0$ and $f(b) > 0$, then the IVM implies there must be some point $c$ between $a$ and $b$ such that $f(c) = 0$. ⟵Section 1.3      Section 1.5 ⟶