# Does continuity imply integrability?

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 .

• The answer depends on what type of integrals you are considering. If you are talking about Riemann integrability on a closed interval $[a,b]$ then any continuous function is integrable. If you are talking about improper Riemann integrals or Lebesgue integrals continuity does not imply integrability. – Kavi Rama Murthy Jun 19 '18 at 7:32
• It depends on the domain too. If the domain isn't compact the integral might not exist. Consider integrating $f(x) = x$ over $\mathbb{R}$. – Shervin Sorouri Jun 19 '18 at 7:34
• Sorry there is a mistake in my comment. $f(x) = x$ is integrable over $\mathbb{R}$ and the value is $0$. Change the domain to $\mathbb{R}^{+}$. – Shervin Sorouri Jun 19 '18 at 7:54
• @ShervinSorouri When I said improper integrals I included integrals over infinite intervals. So your comment is already part of my comment. – Kavi Rama Murthy Jun 19 '18 at 8:31
• Yeah but at the time that you had commented i was writing mine. 2 mins late :( – Shervin Sorouri Jun 19 '18 at 10:32

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 $(0,1)$.