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

Explanation

Given `f(x) = x`
To check continuity of `f(x)`,

We will check it's continuity at anypoint `x=c`


Let c be any real number
`f` is continuous at `x=c`

if `underset(x toc)(lim)f(x)=f(c)`


L.H.S.

`underset(x toc)(lim)f(x)=underset(x toc)(lim)x`

`underset(x toc)(lim)f(x)=c`


R.H.S.

`f(c)=c`


Since, L.H.S.= R.H.S.

`therefore` Function is continuous at `x=c`


Thus, we can write that

`f` is is continuous for `x=c`, where `c` `varepsilon` `RR`

`f` is continuous for every real number.