I know the definition of bounded function i.e
A function that is not bounded is said to be unbounded.
Sometimes, if $f(x) ≤ A$ for all $x \in X$, then the function is said to be bounded above by $A$. On the other hand, if $f(x) ≥ B$ for all $x\in X$, then the function is said to be bounded below by $B$.
And the definition of Boundedness Theorem is a continuous function on a closed bounded interval is bounded and attains its bounds.
But I don't understand in which type of question it is used and how to use it. Please provide me some good level examples.
Here are two important results: (1). Rolle's Theorem: If $f$ is differentiable and $f(a)=f(b)$ with $a<b$ then there exists $c\in(a,b)$ with $f'(c)=0$. ...(2). Intermediate Value Theorem (IVT). If $g$ is differentiable and $a<b$ then there exists $c\in (a,b)$ with $g'(c)=(g(b)-g(a))/(b-a).$.... The proof of (1) uses the continuity of $f$ to show there exists $c\in (a,b)$ satisfying $f(c)=\max \{f(x):x\in [a,b]\}.$.... To prove (2), let $f(x)=g(x) +Kx+L,$ where $K$ and $L$ are constants such that $Ka+L=0$ and $Kb+L=g(a)-g(b).$ So $f(a)=f(b),$ so apply Rolle's Theorem to $f.$ Note that in both theorems that continuity of the derivatives f' or g' is NOT required.