Theorem 1   If $f$ is a continuous complex function on the rectifiable curve $\gamma$ of the complex plane, then

(1)

where $$l = \int_\gamma|dz|$$ is the length of $\gamma$ .

The form of (1) concerning the continuous real function $f$ on the interval $[a,\,b]$ is $$\left|\int_a^b f(x)\,dx\right| \leqq \max_{a\leqq x\leqq b} |f(x)|\cdot(b\!-\!a).$$

For applications of this important theorem, see the example of using residue theorem.