Relation between differentiable,continuous and integrable functions.

Let $g(0)=1$ and $g(x)=0$ for all $x\ne 0$. It is straightforward from the definition of the Riemann integral to prove that $g$ is integrable over any interval, however, $g$ is clearly not continuous.

The conditions of continuity and integrability are very different in flavour. Continuity is something that is extremely sensitive to local and small changes. It's enough to change the value of a continuous function at just one point and it is no longer continuous. Integrability on the other hand is a very robust property. If you make finitely many changes to a function that was integrable, then the new function is still integrable and has the same integral. That is why it is very easy to construct integrable functions that are not continuous.

Probably the simplest example of an integrable function that's not continuous is something like $$f(x)=\left\{\begin{array}{rl}3 & 0\leq x<1\\ 5 & 1\leq x\leq 2.\end{array}\right.$$

This $f$ is clearly not continuous at 1, but it is Riemann integrable on $[0,2]$, with $\int_0^2 f(x)\ dx = 8$.

In order for some function f(x) to be continuous at x = c, then the following two conditions must be true: f(c) is defined and the limit of f(x) as x approaches c is equal to f(c). Recall that the limit of a polynomial p(x) as x approaches c is p(c), therefore polynomials are always continuous.

In order for some function f(x) to be differentiable at x = c, then it must be continuous at x = c and it must not be a corner point (i.e., it's right-side and left-side derivatives must be equal).

Continuity implies integrability; if some function f(x) is continuous on some interval [a,b], then the definite integral from a to b exists. While all continuous functions are integrable, not all integrable functions are continuous. This is because the limit L of some function f(x) as x approaches c can exist despite discontinuity at x = c, so long as f(x) approaches L from both sides of x. Recall that the definition of the definite integral is based upon a Riemann Sum as n tends to infinity (or dx tends to zero, if you prefer) using the limit process. Therefore, any behavior of the function that would cause the limit to not exist would also cause the function to not be integrable. Recall that there are three situations that commonly cause a limit to not exist: different right-side and left-side limits, oscillation, and asymptotes or unbounded/infinite curves.