Question

 Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4. 

Answer 1

7(x) = x-4; x ≥ 4
= 4 - X ;  X<4
`L.H.L : lim_(x->4) f(X)= lim_(x->)4- x =4-4=0`

`R.H.L ; = lim_(x->4) f (X) = lim_(x->4)=4-4=0`

f(4) = 4-4 =0
∴ f(x) is continuous at X = 4.

`R.H.D = f '(4^+) = lim_(h->0)  (f(4+h)-f(4))/((4+h)-4)` 

                           = `lim_(h->0) ((4+h-4)-0)/h`

                           = `lim_(h->0) 1 = 1`

`L.H.D = f'(4^-)=lim_(h->0)  (f(4)-f(4-h))/(4- (4 -h))`

`=lim_(h->0) 0 -(4-(4-h))/h `

`=lim_(h->0)- h/h =-1`

∴ L.H.D . ≠R.H.D.

∴ f '(4) does not exists.
∴ f (4) is continuous at X = 4 but non differentiable at X=4.