True or false? If f is continuous on (a,b), then f(a) and f(b) can be defined so that f isintegrable on [a,b].

Question

True or false? If f is continuous on (a,b), then f(a) and f(b) can be defined so that f is integrable on [a,b].

I know the answer is false but why is it false?

Answer 1

Take $f(x)=\frac1x$ over $]0,1[$. Then, choosing any value for $f$ at $0$ will not change the fact that $f$ is not integrable on ]0,1[.