Suppose the period is T0

Then

f(x+T+1) = f(x+1+T) = f(x+1)

f(x+T+3) = f(x+3+T) = f(x+3)

Hence, f(x+T+1)+f(x+T+3) = f(x+1)+f(x+3) = 2x

However, we also have f(x+T+1)+f(x+T+3) = 2(x+T)

This implies that 2x = 2(x+T) or T = 0

Hence f is non-periodic