# Upper and Lower Bounds of Integral/Corollary

## Corollary to Upper and Lower Bounds of Integral

Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Suppose that $\forall t \in \left[{a \,.\,.\, b}\right]: \left|{f \left({t}\right)}\right| < \kappa$.

Then:

$\displaystyle \forall \xi, x \in \left[{a \,.\,.\,b}\right]: \left|{\int_x^\xi f \left({t}\right) \rd t}\right| < \kappa \left|{x - \xi}\right|$

## Proof

Follows directly from Upper and Lower Bounds of Integral.

$\blacksquare$