Which one of the following is correct in respect of the function `f(x)=(x^(2))/(|x|)" for "xne0andf(0)=0`

Options

A. f(x) is discontinuous every where
B. f(x)= is continuous every where
C. f(x) is continuous at x=0 only
D. f(x) is discontinuous at x=0 only

Answer

B

Explanation

`f(x)={{:((x^(2))/(x)",",xne0),(0,x=0):}`
`={{:((x^(2))/(x)=x",",xgt0),(0",",x=0),(x^(2)/(-x)=-x",",xlt0):}`
Now, `underset(xto0)limf(x)=underset(xto0)lim(-x)=0`
`underset(xto0)limf(x)=underset(xto0)lim(x)=0`
and f(0)=0
So, f(x) is continuous for all other values of x.
Hence, f(x) is continuous everywhere.