= Solution
{c}
The <continuous function> $|h|$ attains a finite <maximum> $M$ on $[a,b]$ by the <extreme value theorem>. Additivity and the <triangle inequality> for the <Riemann integral> imply, for any $x,y\in[a,b]$,
$$
\boxed{|H(y)-H(x)|=\left|\int_x^yh(t)\,dt\right|\leq M|y-x|.}
$$
For reversed endpoints, reverse the orientation of the <integral> before taking its <absolute value>. Given $\varepsilon>0$, choose $\delta=\varepsilon/(M+1)$. Then $|x-y|<\delta$ implies $|H(x)-H(y)|<\varepsilon$. Thus \b[$H$ is <continuous> on $[a,b]$], including one-sided <continuity> at the endpoints; it is even <Lipschitz continuous>. If $a=b$, the <function> is defined on one point and the conclusion is immediate.
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