If $f$ is continuous on [0,$∞$) and $\lim_{x\to\infty} f(x)= 0$, then prove that $\int_{0}^{∞} f'(x) dx = -f(0).$

Question

If $f$ is continuous on [0,$∞$) and $\lim_{x\to\infty} f(x)= 0$, then prove that $$\int_{0}^{∞} f'(x) dx = -f(0).$$

Any hints?

Answer 1

If $f'$ is not continuous, then $\int_{0}^{x}f'$ need not exist for all $x \geq 0$; hence I assume the hypothesis should be $f$ is continuously differentiable, rather than simply continuous.

If $f$ is continuously differentiable on $]0, \infty[$, then $\int_{0}^{x}f'$ exists and $\int_{0}^{x}f' = f(x) - f(0)$ for all $x \geq 0$ by the fundamental theorem of calculus; hence $$ \int_{0}^{\infty}f' = \lim_{x \to \infty}\int_{0}^{x}f' = \lim_{x \to \infty}\Big[ f(x) - f(0) \Big], $$ which $\to -f(0)$ by the assumption that $f(x) \to 0$ as $x \to \infty$.