Can somebody please explain , in simple terms , if continuity ALWAYS imply integrability ? If it doesn't ? Maybe some counter examples?
Or maybe even what are the necessary conditions in order to imply that a continuous function can be integrable?
Moreover, i wanted to be sure that differentiabilility always implies continuity … is this correct? I mean if i say that f is differentiable on point a then f is continuous on point a . is that wrong to say ? Or may it be right to say that if a left and right derivative of a fucntion exist at a point a , then there exists the left and right continuity at point a. Left and right continuity at point a together imply continuity .
Best Answer
This answer refers to single variable Riemann integrability. It is easier to define integrability for bounded functions and due to Weierstrass' theorem, a continuous function on a closed interval is bounded. Indeed any continuous function on a closed interval is integrable (but not any bounded function on a closed interval: for example, Dirichlet function = indicator of rational numbers, isn't integrable). However, not any continuous function on an open interval is integrable; For example take $1/x$ in $(0,1)$.