Let’s consider the function $\displaystyle f(x) = \frac{e^x - 1}{x}$ from the post about limits and defined a piecewise function $g$, where $g(x) = f(x)$ if $x \neq 0$ and $g(0) = 1$.

Notice that unlike before, $g$ is well-defined everywhere since we simply define the value of $g$, where $f$ was undefined. Additionally, we now have the case that the limit of the function as we approach $x = 0$ is $1$. That is,

$$\lim_{x\to 0} g(x) = 1 = g(0).$$

In this case, we say that $g$ is continuous at $x = 0$. In fact, $g$ is continuous at every point in its domain, so we say that $g$ is a continuous function

Definition. a function $f$ is continuous at $x = a$ if $\lim_{x\to a} f(x) = f(a)$. If $f$ is continuous over its entire domain, it is a continuous function.

Examples

1. Polynomial functions $f(x) = a_0 + a_1x + \cdots + a_n x^{n}$ are continuous.

2. Trig functions $\sin(x)$, $\cos(x)$ are continuous.

3. $f(x) = a^x$ is continuous.

4. A function defined by the product of any 2 continuous functions is continuous.

5. A function defined by the sum of any 2 continuous functions are continuous.