Let $V$ be the vector space of all functions from $\mathbb R$ into $\mathbb R$ which are continuous. Let $T$ be the linear operator on $V$ defined by $$(Tf)(x) = \int_0^x f(t) dt$$ Prove that $T$ has no eigen values.
All those who are going to differentiate in the middle, please consider, there are functions which are continuous, but nowhere differentiable. Ex: Weirstrauss function. So, you can't differentiate anywhere in between.