Let $\lambda \in \mathbb{R}$ be a constant, $\lambda \neq 0$. Is there a function $f \in C[0,1]$, $f \neq 0$, that satisfies the following relation:
$\lambda f(s) = \int_0^s f(t) \, dt$
Attempt at a solution:
Applying Fundamental Theorem of Calculus, one gets $ \lambda f(s) = F(s) - F(0)$, which implies that $f(0) = 0$. It follows that $f$ cannot be differentiable, because if it were, then taking the dervative on both sides gives $\lambda f'(s) = f(s)$, which gives together with the boundary condition $f(s) = \exp[\frac{s}{\lambda}] - 1$, but this function does not satisfy the relation. Hence if such a function exists, then it cannot be differentiable on $[0,1]$.