Show that the function f x = modulus x+2 modulus is continuous at every x E R but fails to be differentiable

at x = -2 .

For x > –2
f(x) = (x + 2). This is a linear function so will be continuous for x > –2.
For x = –2
f(x) = 4
For x < –2
f(x) = –(x + 2). This is a linear function so will be continuous for x < –2.

LHL at x = –2

RHL at x = –2

Hence at x = 4
LHL = RHL = f(4) = –4.
Hence, the function is continuous at x = –4 and it is also continuous at every

Checking the differentiability:
For x > –2

For x < –2

Since the above two are not equal, the function is not differentiable at x = –2.