Question

How do you find the points of continuity of a function?

Answer 1

Hint: We first discuss the concept of the continuity for a function $f\left(x\right)$ at a particular point $x=a$. We try to find the limiting value. We also use examples to understand the concept better. We find the relation between continuity and the differentiation.

Complete step by step answer:
A given function $f\left(x\right)$ is continuous if the limiting value of the function at a particular point is equal from both ends.
This means if we have to check the continuity of the function $f\left(x\right)$ at point $x=a$ then we have to find the value of the function at three parts $x=a^{+},a^{-},a$.
If the equation $\underset{x\to a^{+}}{lim}f\left(x\right)=\underset{x\to a^{-}}{lim}f\left(x\right)=f\left(a\right)$ holds then we can say that the function is continuous at $x=a$.
We take two functions to understand the theorem better.
Let $f\left(x\right)=\left|x\right|$ and we find continuity at $x=0$.
Now $\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0}{lim}x=0$, $\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0}{lim}(-x)=0$ and $f\left(0\right)=0$.
Therefore, $f\left(x\right)=\left|x\right|$ is continuous at $x=0$.
Now we take $f\left(x\right)={\displaystyle \frac{1}{x}}$ and we find continuity at $x=0$.
Now $\underset{x\to 0^{+}}{lim}{\displaystyle \frac{1}{x}}=+\mathrm{\infty }$, $\underset{x\to 0^{-}}{lim}{\displaystyle \frac{1}{x}}=-\mathrm{\infty }$ and $f\left(0\right)=\text{undefined}$.
Therefore, $f\left(x\right)={\displaystyle \frac{1}{x}}$ is not continuous at $x=0$.

Note: The differentiation of a function is connected to its continuity where if a function is differentiable then it is definitely continuous. But the opposite is not always true. A function being continuous doesn’t make it differentiable.