What is continuity?

Think of it as being able to draw a graph with a pencil without taking your pencil off the paper. If you can do so, then the graph is continous, otherwise it is discontinuous.

Example of a continuous graph

Following the cursor, it smoothly traces over the function $f(x) = \sin(x)$, indicating that it is continuous over the domain $[0,\ 2\pi]$.

In fact, $\sin(x)$ is continuous for all $[-\infty, \ +\infty]$.

Example of a discontinuous graph

In this animation, you can see how the cursor must "jump" from on part of the function to another, which makes it discontinuous.

However, do note that the function is continuous from $0$ to $2$ non-inclusive, or $[0, \ 2)$, and from $2$ inclusive to $+\infty$, or $[2,\ +\infty)$.

Types of discontinuities

There are 3 types of discontinuities that you should be aware of:

Determining continuity at a point

To determine whether or not a point on a is continuous at some $x=a$, the following must be true:

$L = \ _{x \to a^{+}}^{\lim}f(x) = \ _{x \to a^{-}}^{\lim}f(x) = f(a) \quad$ or $\quad \ _{x \to a}^{\lim}f(x) = f(a)$

That is, the limit of the function at $x=a$ is equal to $f(a)$

Example

Determine whether the function $f(x) = \left\{\begin{matrix} 3(x-2)^{2}+1 \quad \textrm{for}\ x<2 \quad (1) \newline 3(x-2)+1 \quad \textrm{for}\ x\geq 2 \quad (2) \end{matrix}\right.$ is continuous or not at $x = 2$

answer

To prove that $f(2)$ is continuous, we must show that $\ _{x \to 2}^{\lim}f(x) = f(2)$

1. Obtain the value of $f(2)$. The sub-function $(2)$ will be used to calculate this value:

$f(2) = 3((2) - 2) + 1 = 3(0) + 1 = 1$

2. Find the left limit of $f(x)$ at $x=2$:

$_{x \to 2^{-}}^{\lim}f(x)$

$= \ _{x \to 2}^{\lim} (3(x-2)^{2}+1)$

$= \ _{x \to 2}^{\lim} (3x^2 - 12x + 13)$

$= 3(2)^2 - 12(2) + 13$

$= 12 - 24 + 13$

$= 1$

3. Find the right limit of $f(x)$ at $x=2$:

$_{x \to 2^{+}}^{\lim}f(x)$

$= \ _{x \to 2}^{\lim} (3(x-2)+1)$

$= \ _{x \to 2}^{\lim} (3x-5)$

$= 3(2)-5$

$= 6-5$

$= 1$

Since $\ _{x \to 2^{+}}^{\lim}f(x) = \ _{x \to 2^{-}}^{\lim}f(x) = f(2) = 1$, $f(x)$ is continuous at $x=2$.

Confirmation

Just to confirm, you can see that in fact, $f(x)$ is continuous at $x=2$ from the image below: