UPSC NDA Math Model Paper 8

A function is differentiable only if it is continuous function for all values of x.
Check whether the given function are continuous at all values of x
Consider the function f(x)=

x

|x|

f(x) is not a continuous function at x = 0.
f(x) is not differentiable function.
Now consider the function f(x) = x|x|
f(x)=x2,x≥0
f(x)=−x2,x<0
f(x) is continuous function at all real values of x
f(x) is differentiable function.
f'(x) = 2x , x ≥ 0 and
f(x) = - 2x , x < 0
Consider the function f(x) =

1

|x|

f(x) is not a continuous function at x = 0.
f(x) is not differentiable function.
Consider the function f(x)=

1

x

f(x) is not a continuous function at x = 0.
f(x) is not differentiable function.