Discuss the continuity of the function f given by `f(x) = |x|` at `x = 0`.

Explanation

`f(x) = |x|`

`f(x)={-x, if x<0; x, if x>=0:}`

`f` is continuous at `x = 0`
if L.H.L.=R.H.L.=`f(0)`

i.e. `underset(x to0^-)(lim)f(x)=underset(x to0^+)(lim)f(x)=f(0)`

Finding L.H.L. and R.H.L.

L.H.L. at `x to 0`

`underset(x to0^-)(lim)f(x)=underset(h to0)(lim)f(0-h)`

`underset(x to0^-)(lim)f(x)=underset(h to0)(lim)f(-h)`

`underset(x to0^-)(lim)f(x)=underset(h to0)(lim)|-h|`

`underset(x to0^-)(lim)f(x)=underset(h to0)(lim)h`

`underset(x to0^-)(lim)f(x)=0`


R.H.L. at `x to 0`

`underset(x to0^+)(lim)f(x)=underset(h to0)(lim)f(0+h)`

`underset(x to0^+)(lim)f(x)=underset(h to0)(lim)f(h)`

`underset(x to0^+)(lim)f(x)=underset(h to0)(lim)|h|`

`underset(x to0^+)(lim)f(x)=underset(h to0)(lim)h`

`underset(x to0^+)(lim)f(x)=0`

And
`f(0)=0`


So, L.H.L.=R.H.L.=`f(0)`

Hence, `f` is continuous at `x = 0`