= Solution
{c}
By the <Heine-Cantor theorem>, a <continuous function> on the <compact interval> $[a,b]$ is <uniformly continuous>. Given $\varepsilon>0$, choose $\delta>0$ such that $|f(s)-f(t)|<\varepsilon$ whenever $s,t\in[a,b]$ and $|s-t|<\delta$.
Choose a finite <partition of an interval> $a=t_0<t_1<\cdots<t_m=b$ with every width below $\delta$. Define the <step function>
$$
\phi(t)=f(t_{j-1})\quad(t_{j-1}\leq t<t_j),\qquad\phi(b)=f(b).
$$
For each $t<b$, its distance from the chosen left endpoint is below $\delta$, so $|f(t)-\phi(t)|<\varepsilon$; at $b$ the error is zero. Hence
$$
\boxed{|f(t)-\phi(t)|<\varepsilon\quad\text{for every }t\in[a,b].}
$$
A <step function> may have separately assigned values at its finitely many partition points; these values do not affect its <Riemann integral>. If $a=b$, simply take $\phi=f$ on that singleton.
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