Answer
No.
Work Step by Step
$\lim\limits_{x \to 0^{-}}f(x)\gt 1$
$\lim\limits_{x \to 0^{+}}f(x)\lt 0$
$\implies \lim\limits_{x \to 0^{-}}f(x)\ne\lim\limits_{x \to 0^{+}}f(x)$
One of the conditions for the function to be continuous at 0 is the existence of the limit at 0. As the left-hand limit and right-hand limit doesn't coincide, $\lim\limits_{x \to 0}f(x)$ doesn't exist. Therefore, the function $f$ can't be continuous.
