Continuity at a Point

Problem 232: Determine whether the following functions are continuous at $a$ .

(1) $\begin{equation*} f(x) = \frac{3x^2 + 2x + 1}{x-1}, \quad \text{at} \quad a = 1\end{equation*}$

Solution: Before we determine whether the given function is continuous or not, we need to know when is a function continuous. That is,

Theorem 1: A function $f$ is continuous at $a$ if

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

Using Theorem 1, one can now determine whether (1) is continuous or not. Notice that

(3) $\begin{equation*} f(1) = \frac{3(1)^2 + 2(1) + 1}{1-1}=\frac{6}{0}=\text{undefined!}\end{equation*}$

Because $f(1)$ is undefined, by Theorem 1 the function $f$ is not continuous at 1.