- #1
evinda
Gold Member
MHB
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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)
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)
