# How to solve this? Determine the continuity and derivability domain for #f(x)# #f:[0,oo)->RR,f(x)=|x-1|sqrtx #

##### 1 Answer

Mar 29, 2017

is continuous in

#### Explanation:

We have that:

so:

- For
#x in [0,1)# ,#f(x) = (-x+1)sqrt(x)# - For
#x = 1# ,#f(x) = 0# - For
#x in (1,+oo)# ,#f(x) = (x-1)sqrt(x)#

We can deduce that the function is continuous in

For

which implies:

so the function is continuous also in

Differentiating

- For
#x in [0,1)# ,#f'(x) = d/dx((-x+1)sqrt(x)) = -sqrtx -(x+1)/(2sqrt(x)) = (1-3x)/(2sqrtx)# - For
#x in (1,+oo)# ,#f'(x) = d/dx((x-1)sqrt(x)) = sqrt(x) + (x-1)/(2sqrtx) =(3x-1)/(2sqrtx)#

so for

As the derivative is not continuous in