Answer
Please see below.
Work Step by Step
As we see the graph of $f(x)=\begin{cases}\frac{\cos x -1}{x}, & x<0 \\ 5x, & x \ge 0 \end{cases}$, we find that the function is continuous everywhere since for all $x \neq 0$ the functions $5x$ and $\frac{\cos x -1}{x}$, being the ratio of two continuous functions with nonzero denominator, are continuous and at $x=0$ we have$$\lim_{x \to 0^-}f(x)= \lim_{x\to 0^+}f(x)=f(0)=0.$$
