Question

The function f(x) = $|x+1|$ on the interval [-2,0] is

• A. continuous and differentiable
• B. Continuous on the integers but not differentiable at all points
• C. neither continuous nor differentiable
• D. differentiable but not continuous

Answer 1

The correct option is B Continuous on the integers but not differentiable at all points

f(x) = $|x+1|$

= -(x+1) for x < -1

= (x+1) for x $\ge$ -1

Only concern is x = -1

Left limit = ${lim}_{x\to -1^{-}}-(x+1)=0$

Right limit = ${lim}_{x\to -1^{+}}(x+1)=0$

f(x) is continuous in the interval [-2,0]

LD = -1

Right Derivative = + 1

Hence f(x) is not differentable at x = -1

Alternative Solution :

It is clear from graph f(x) is continuous every where but not differentiate at x = -1 because there is sharp change in slope at x = -1.