Roots along a homotopy

Question

Suppose we have two functions $f,g:\mathbb{R}^n\rightarrow \mathbb{R}$. Assume the existence of unique roots to both function, i.e. $x^f,x^g\in \mathbb{R}^n$ such that $f(x^f)=0=g(x^g)$. Define the homotopy $$H(x,t)= tf(x)+(1-t)g(x) .$$ Are there any known conditions such that $H(\cdot,t)$ has a unique root for every $t$? This is not my field. I have found this example which seems to suggest that some non-trivial conditions have to be made. I am fine with any regularity assumptions (continuity, smoothness, ...) but judging by the example with polynomials, this is not enough. I am also fine with assuming that $f,g$ take values in a bounded interval (which includes 0) if that helps.