UPSC NDA Math Model Paper 7

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.