Answer
Yes.
Work Step by Step
We know that if a function is odd, then $f(-x)=-f(x).$
We know that if a function is even, then $f(-x)=f(x).$
If $f(x)=0$, then $f(-x)=0=-f(x)=f(x)$, thus in this case the function is both odd and even, thus the answer is yes.
