Counter example

Question

I know that for an operator $K:L^2(0,1)\rightarrow L^2(0,1)$ defined by

$$K\phi(x)=\int_0^{x}\phi(t)dt$$ is not onto.

Can someone give an example of an element which belongs to co-domain but not in range.

Answer 1

The functions in the range of $K$ are always continuous so any $g \in L^2(0,1)$ which does not equal a.e to a continuous function will give you an example. For a concrete example, take $g$ to the characteristic function of any closed interval contained in $(0,1)$.