TITLE: How does one show that a function is continuous over some interval?
QUESTION [8 upvotes]: As title says, how does one show that a function is continuous over some interval (let us say over some interval of real numbers?)
Would(Can) this involve derivative?

REPLY [7 votes]: If you know that a given function $f$ is differentiable on some interval $(a, b) \subseteq \mathbb R$, then it certainly follows that $f$ is continuous on that interval (since differentiability requires continuity and more). 
But a function $f$ can be continuous on an interval, even though it may not be differentiable on the entire interval. In such cases, and in general, we have that:

$f(x)$ is continuous on $[a,b]$ if and only if the following holds: 

$\forall x_0 \in (a,b) \subseteq \mathbb R,\;\; \lim_{x \to x_0} f(x) = f(x_0);$
$\lim_{x\to a^+}f(x)=f(a)\;\;\text{and}\;\;\lim_{x \to b^-} f(x) = f(b).$