Given: f(x)=|x|+3 The given function can be re-written as: f(x)={
3−x,
x<0
x+3,
x≥0
Let's examine the continuity of f(x) at x = 0 ⇒
lim
x→0−
f(x)=
lim
x→0
[3−(0−h)]=3 ⇒
lim
x→0+
f(x)=
lim
x→0
[(h+0)+3]=3 ⇒
lim
x→0
f(x)=3=f(0)=3 So, the given function f(x) is continuous on R. Similarly, we can see that f(x) is differentiable over R\{0} Hence, option D is the correct answer.