Limit

In general, $h(x)$ is continuous on a given closed interval $[a,b]$ if and only if

• (i) $\forall x_{0} \in (a,b), \displaystyle\lim_{x \to x_{0}} h(x) = h(x_{0})$ , and

• (ii) $\displaystyle\lim_{x \to a^{+}} h(x) = h(a)$ and $\displaystyle\lim_{x \to b^{-}} h(x) = h(b)$ .

As conditions (i) and (ii) both hold for the given function $h(x)$ on $[a,b]$ we can conclude that $\boxed{Yes}$ , $h(x)$ is continuous on the interval $[a,b]$ .