How does one show that a function is continuous over some interval? 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?
 A: 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:


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


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*$\forall x_0 \in (a,b) \subseteq \mathbb R,\;\; \lim_{x \to x_0} f(x) = f(x_0);$

*$\lim_{x\to a^+}f(x)=f(a)\;\;\text{and}\;\;\lim_{x \to b^-} f(x) = f(b).$


A: Basically, you have to show that:


*

*$f(x)$ is continuous on $(a,b)$, i.e; $\forall x_0\in(a,b), \lim_{x\to x_0}f(x)=f(x_0)$

*$\lim_{x\to a^+}f(x)=f(a),~~\lim_{x\to b^-}f(x)=f(b)$
I assume you aregoing to show that if $f(x)$ is continuous on $[a,b]\subset\mathbb R$.
A: Continuity has various equivalent definitions.  The existence of a derivative is a stronger property than continuity, so if you already know the function is differentiable on an interval, then it is certainly continuous, but the converse does not hold.
