Question

If f is continuous on [a, b] and integral a to b of f(x)g(x)dx=0 for all continuous functions g on [a, b] then f is identically equal to 0 on [a, b]

Answer 1

Conditions

If $f$ is continuous on $[a, b]$ and integral $a$ to $b$ of $f(x)g(x)dx=0$ for all continuous functions $g$ on $[a, b]$ then $f$ is identically equal to 0 on $[a, b]$

Solution

This is not true. Consider the counterexample:

$$f(x) = \sin x$$ $$g(x) = 1$$ $$x \in [a, b] = [0, 2\pi]$$

Then $f, g$ are continuous on $[a, b]$ and

$$\int_{0}^{2\pi} \sin x \, dx = 0$$

but

$$f(x) = \sin x \neq 0 \quad \forall x \in [0, 2\pi]$$