Measure, Integration & Real Analysis · Problem 1A.11

The key insight is that Riemann integrability gives you uniform control over how much $f$ can accumulate over small intervals—this is exactly what you need to make $F(t)$ change arbitrarily little when $t$ moves slightly. Rather than tracking $f$'s behavior pointwise, use the definition of Riemann integrability to bound $\left|\int_s^t f\right|$ for nearby points $s$ and $t$, which directly translates to $|F(t) - F(s)|$ being small. Continuity at $a$ and interior points follow from the same principle.

The key insight is that Riemann integrability gives you uniform control over how much $f$ can accumulate over small intervals—this is exactly what you need to make $F(t)$ change arbitrarily little when $t$ moves slightly. Rather than tracking $f$'s behavior pointwise, use the definition of Riemann integrability to bound $\left|\int_s^t f\right|$ for nearby points $s$ and $t$, which directly translates to $|F(t) - F(s)|$ being small. Continuity at $a$ and interior points follow from the same principle.