find the number of continious function?

Question

how many number of continious function on R which satisfy $ [f(x)]^2 =x^2$ for all $x\in R$ ?

1) 2

2) 4

3) 1

4) 6

i think answer will be option 1 becoz $x^2= 1$ has two solution

Answer 1

There are four solutions.

1) y=x

2) y=-x

3) y=|x|

4) y= -|x|

Notice that these are continuous functions and they are different.