Prove that the function ƒ given ƒ(x)=|x-3|, x є R is continuous but not differentiable at x=3

f(x)=|x-3|

Since every modulus function is continuous for all real x, f(x) is continuous at x=3.

f(x) =

To prove differentiable , we will use the following formula.

= = f(a)

L.H.L

=

=

= 1

R.H.L:

=

=

= -1

Since, L.H.L R.H.L, f(x) is not differentiable at x=5.