Question

Is $f$ continuous at $0$?

Short answer

$f\left(x\right)$is not continuous at$x=0$.

Step-by-step solution

Step 1. Graph the function f(x)

Let us graph $f\left(x\right)$with the help of a graphing utility.

Step 2. Determine f(x) is continuous at 0

For a function to be continuous at$x=c$, then the function must be defined at $x=c$and$\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=f\left(c\right)$.

Let us consider the behavior of $f\left(x\right)$near $x=0$.

When we consider the limit coming from the left side, the function approaches a value of $4$.

That is$\underset{x\to 0^{-}}{\lim }f\left(x\right)=4$

When we consider the limit coming from the right side, the function approaches a value of $1$.

That is$\underset{x\to 0^{+}}{\lim }f\left(x\right)=1$

Since$\underset{x\to 0^{-}}{\lim }f\left(x\right)\ne \underset{x\to 0^{+}}{\lim }f\left(x\right)$,$f\left(x\right)$is not continuous at$x=0$.