Question

is a differentiable function is continuos or Riemann integrable
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Answer 1

I have been doing lots of calculus these days and i want to confirm with you guys my understanding of an important concept of calculus.

Basically, in the initial phase,students assume that integration and differentiation are always associated to each other, i.e., a function which is integrable is also differentiable at the same time. But having explored on it more, i found out its not true at all and does not hold always. Many a times (or should i say infinitely times) a function can be integrable on an interval while its not differentiable on that same interval (and vice versa) .

What i want to ask is this : i recently read a conclusion on the above mentioned concept which is :

{Differentiable functions} ⊂ {Continuous functions} ⊂ {Integrable functions}

i.e., each is a proper subset of the next. Now, "Differentiable functions set" is a proper subset of "Continuous functions set"... that is very well understood without a doubt as every continuous function may or may not be differentiable. I have problem with the next relation which is :

"Continuous functions set" is a proper subset of "Integrable functions set"...Why is this so???

I am just not able to visualize this. I know that a bounded continuous function on a closed interval is integrable, well and fine, but there are unbounded continuous functions too with domain R , which we cant say will be integrable or not.