Estimation lemma

Statement

Let $\gamma : [a, b] \to \mathbb{C}$ be a continuous and piecewise continuously differentiable path with length $L(\gamma) = \int_a^b |\gamma'(t)| \, dt$. Let $f$ be a complex function that is continuous on the image of $\gamma$. Let $M \ge 0$ be a real number with $|f(z)| \le M$ for each $z$ on the image of $\gamma$. Then the contour integral of $f$ along $\gamma$ satisfies $$\left| \int_\gamma f(z) \, dz \right| \le M \, L(\gamma).$$