Question

Prove that the identity function on real numbers given by $f\left(x\right)=x$ is continuous at every real number.

Answer 1

Given: function is $f\left(x\right)=x$

Now we check the continuity at $x=a(a$ is any real constant)

If $f\left(x\right)$ is continuous at $x=a$ then
$\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)=f\left(a\right)$

Finding L.H.L.

$\underset{x\to a^{-}}{lim}x$

$=\underset{h\to 0}{lim}(a-h)$

Putting $h=0$ then we get,
$=(a-0)=a$

Finding R.H.L.

$\underset{x\to a^{+}}{lim}x$

$=\underset{h\to 0}{lim}(a+h)$

Putting $h=0$ then we get,
$=\underset{h\to 0}{lim}(a+0)=a$

To find $f\left(x\right)$ at $x=a$
$f\left(x\right)=x\text{at}x=a$
$f\left(a\right)=a$

Hence, $\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)=f\left(a\right)$
Therefore, the function $f\left(x\right)=x\text{is continuous at}x=a$