I know that a function is continuous at a point if the limit from left and right side exists and are equal and for a function to be continuous, the function should be continuous at all points. My question is that if I want to check continuity of a function, I cannot practically check continuity at each and every point in $\mathbb{R}$. So, how to do it?
Check continuity of a function
calculuscontinuity
Related Solutions
As Jose27 noted, uniformly continuous functions need not be differentiable even at a single point.
It is true that if $f$ is defined on an interval in $\mathbb R$ and is everywhere differentiable with bounded derivative, then $f$ is uniformly continuous. In fact, it follows from the Mean Value Theorem that such an $f$ is Lipschitz, which is much stronger.
However, if $f$ is uniformly continuous and everywhere differentiable, then $f$ need not have bounded derivative. Jose27 mentions $\sqrt x$, which would work as an example on the interval $(0,\infty)$. The function $$f\left(x\right)=\begin{cases}x^2\sin\left(\frac{1}{x^2}\right) & :x\neq 0\\ 0 &:x=0\end{cases}$$ is uniformly continuous on any bounded interval such as $(-1,1)$, but has unbounded derivative near $0$. There are also examples where $f'$ is bounded on bounded intervals, but unbounded on $\mathbb R$, while $f$ is uniformly continuous. You can show that any continuous function $f$ on $\mathbb R$ such that $\lim\limits_{|x|\to \infty}f(x)=0$ is uniformly continuous, and using this fact you can see that Nate Eldredge's example here of $\sin(x^4)/(1+x^2)$ provides such an example. Another source of examples is the question
Why if $f'$ is unbounded, then $f$ isn't uniformly continuous?
The real question is, why is the $0<d_X(x,p)$ condition in the definition of the limit?
Basically, it doesn't hurt to allow $x=p$ in the continuity example, because (1) we know that $p\in E$, and (2), when $x=p$, $d_Y(f(x),f(p))=0<\epsilon$, so there is no reason to leave it out.
Essentially, in the continuity case, the case $x=p$ is trivially true.
On the other hand, when $p\in E$ in the limit definition, we don't want the limit to depend on $f(p)$, but only on the values $F(x)$ when $x\neq p$.
For example, when $f(x)=0$ for $x\neq p$ and $f(p)=1$, we want $\lim_{x\to p} f(x) = 0$, but that would not be true if we didn't have the condition $0<d_X(x,p)$ - without that condition, the limit is undefined.
Isolated points:
We consider isolated points to be points of continuity because we want in general that if:
$f:E\to Y$ is continuous at $p$ and $p\in E'\subset E$ then $f_{|E'}:E'\to Y$ to also be continuous at $p$.
However, note that the continuity definition could have said $0<D_X(x,p)$. That doesn't affect the continuity at $p$ one bit, so if you wanted to define all isolated points as points of discontinuity, you'd want some other definition completely.
Best Answer
Well, that is not the rigurous definition of continuity but it works for most UNDERGRADUATE functions.
Some tricks to check continuity:
These are just a few tricks; they won’t prove continuity in every case, but for undegraduate students they may be enough.