Answer
(a) 0,
(b) 0,
(c) 0,
(d) $f$ is continuous at $x=0$
Work Step by Step
Given
$$f(x)=\left\{\begin{array}{ll}{2 x^{2}} & {\text { when } x<0} \\ {4 x} & {\text { when } x \geq 0}\end{array}\right.$$
(a) Since
\begin{align*}
\lim _{x \rightarrow0^{-}} f(x)&=\lim _{x \rightarrow0^{-}}2 x^2\\
&=\lim _{x \rightarrow0^{-}}2(0)\\
&=0
\end{align*}
(b) Since
\begin{align*}
\lim _{x \rightarrow0^{+}} f(x)&=\lim _{x \rightarrow0^{+}} (4x)\\
&=4(0)\\
&=0
\end{align*}
(c) $f(0)= 4(0)=0$
(d) Since $$\lim _{x \rightarrow0^{+}} f(x)=\lim _{x \rightarrow0^{-}} f(x)=f(0)=0$$
Thus, $f$ is continuous at $x=0$
