Lebesgue differentiation theorem with two variables

Question

We know that if $f:\mathbb{R}\to\mathbb{R}$ is continuous then $$\lim_{h\to 0}\frac{1}{h}\int_x^{x+h} f(s)ds=f(x).$$ But if we have $f:\mathbb{R}\times \mathbb{R}\to\mathbb{R}$, what kind of dependence in the second variable we should assume to have: $$\lim_{h\to 0}\frac{1}{h}\int_x^{x+h} f(s,h)ds=f(x,0)\ \ \ \ ?$$