Suppose $f : [a,b] \to \mathbb{R}$ is a continuous function. Can another continuous function $g : [a,c] \to \mathbb{R}$ be defined such that $\int_a^b f(x)dx =\int_a^c g(x)dx$ so that $f(x) = g(x), x\in[a,b]$ and $c>b$
(Note: if $g \geq 0$ on $[b,c]$ this will not be possible but this restriction is not imposed on the above.)