multivariable calculus – should the function with the following property be vanish at some point?

Supose $f: mathbb{R}^2 rightarrow mathbb{R}$ smooth such that $f(x+1,-y) = -f(x,y)$, should the function with this property be vanish at some point?

I’ve tried direct inspection, tried to override one of the coordinates to see if the result was an odd function, but nothing seems to work. Is there any other way to address this problem?