#### Question

Given f (x) = 2x, x < 0

= 0, x ≥ 0

then f (x) is _______.

(A) discontinuous and not differentiable at x = 0

(B) continuous and differentiable at x = 0

(C) discontinuous and differentiable at x = 0

(D) continuous and not differentiable at x = 0

#### Solution

(D) continuous and not differentiable at x = 0

solution:

f (x) = 2x, x < 0

= 0, x ≥ 0

`lim_(x->0^-)f(x)=lim_(x->0^-)2x=0`

`lim_(x->0^+)f(x)=lim_(x->0^+)0=0`

and f(0) = 0

`lim_(x->0^-)f(x)=lim_(x->0^+)f(x)=f(0)`

Hence, f(x) is continuous at x = 0.

Now we find left hand derivative and right hand derivative of f(0) at x = 0

Right hand derivative at x = 0

i.e `f'(0^+)=lim_(h->0^+)(f(0+h)-f(0))/h=lim_(h->0^+)(0-0)/h=0`

Left hand derivative at x = 0

i.e `f'(0^-)=lim_(h->0^-)(f(0+h)-f(0))/h=lim_(h->0^+)(h-0)/h=1`

`f'(0^+)ne f'(0^-)` Hence, f(x) is not differentiable at x = 0.