Find Continuous Functions Subject to an Integral Condition

[evinda]

Hello! (Wave)

I want to find all the continuous functions $f: [0,1] \to \mathbb{R}$ for which it holds that:$$\int_0^1 f(t) \phi''(t) dt=0, \forall \phi \in C_0^{\infty}(0,1)$$

If we knew that $f$ was twice differentiable, we could say that $\int_0^1 f(t) \phi''(t) dt= \int_0^1 f''(t) \phi(t) dt+ f(1) \phi'(1)-f(0) \phi'(0)$

What can we do if we are not allowed to differentiate $f$ ? (Thinking)

 

[Ackbach]

Try the Fundamental Lemma of the Calculus of Variations.