Continuity and Differentiability - Online Test

Case 1 Let c be a real number which is not equal to any integer. for all real numbers close to c the value of the function is equal to [c]; i.e., limx→cf(x)=limx→c[x]=[c]. Also f(c)=[c]and hence the function is continuous at all real numbers not equal to integers.

Case 2 Let c be an integer. Then we can find a sufficiently small real number r>0 such that [c−r]=c−1whereas[c+r]=c

This, in terms of limits mean that limx→c−f(x)=c−1,limx→c+f(x)=c

Since these limits cannot be equal to each other for any c, the function is discontinuous at every integral point.

Here,f(x)=x|x,|whichisproductoftwocontinuousfunctions,Hencefiscontinuous∀x∈R

Also, f'(x) =x.x|x|+|x|.1
=|x+||x|=2|x|,

Therefore f'(x) exists at all x∈R

Further, f'(0) = 2.|0|=0