Does continuity imply integrability?

Question

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 .

Answer 1

I found this, slightly technical, definition, which should be relevant:

What is Riemann integrable function?

A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure)

Secondly, differentiability implies continuity at a point. See this

Answer 2

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 $(0,1)$.