How to check continuity of a function? 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?
 A: Well, that is not the rigurous definition of continuity but it works for most UNDERGRADUATE functions.
Some tricks to check continuity:

*

*You may know that elementary fucntions ($\sin$, $\cos$, $\exp$, polynomials,...) are continuous everywhere.

*$\log(\cdot)$ is continuous on its domain (whenever what is inside is strictly positive).

*$\sqrt{\cdot}$ is continuous on its domain (wheneever what is inside is positive (not stricly).

*Moreover, any linear combination of continuous functions is also continuous (say, the addition of subtraction, and multiplication by a number). Also multiplication of continuous functions is also continuous.

*For quotient of contuinuous functions, everything works okay EXECEPT for those points that cancel the denominator.

These are just a few tricks; they won’t prove continuity in every case, but for undegraduate students they may be enough.
A: One way is to take an arbitrary point $x_0$ with no prior assumptions and show that for every $\varepsilon>0$ there exists $\delta>0$ (which may depend on $x_0$) s.t for every $x\in (x_0-\delta,x_0+\delta)$, $|f(x)-f(x_0)|<\varepsilon$. This is the case in $\Bbb{R}$, but this is the basic definition.
For example if we wanted to show that the function $f(x)=x$ is continuous in $\Bbb{R}$, we may say the following: let $x_0\in\Bbb{R}$, $\varepsilon>0$. So we take $\delta=\varepsilon$, so for every $x$ s.t $|x-x_0|<\delta$ we get $|f(x)-f(x_0)|=|x-x_0|<\delta=\varepsilon$, so $x$ is continuous in $x_0$ and therefore because we made no prior assumptions, it's continuous in $\Bbb{R}$.
A: There are several „methods“ to check continuity of a function $f:\Bbb R \longrightarrow \mathbb{R}$:

*

*show that given an arbitrary point $x$ and any sequence $x_n \rightarrow x$ converging to $x$ you have that $f(x_n) \rightarrow f(x)$. This is feasible, if your function itself is given by a formula closely related to limits, like $\exp, \sin, \cos, x\mapsto x^2$ etc. Here you may use facts like (under certain hypotheses) limits commute or are compatible with multiplication, addition etc.

*show that it is given by a composition of continuous functions. For example $x \mapsto e^{x^2}$ is continuous due to $x \mapsto e^x$ and $x\mapsto x^2$ being continuous.

*show that it is induced by some sort of universal mapping property, by which I mean some sort of general theorem telling you that for example having continuous functions $f:\Bbb R \rightarrow \Bbb R, g: \Bbb R \rightarrow \Bbb R$ induces a unique continuous function $(f,g):\Bbb R \rightarrow \Bbb R^2, x \mapsto (f(x),g(x))$.

