Marginal Distributions Are Point Mass

Question

Let $f(x,y,z)$ be some unknown joint density of the random variables $X,Y,Z$. What can I say about the joint density if I know that the marginal distribution of $Y$ and $Z$ are pretty much a point mass? I. e.: $$\int_{\mathcal X}\int_{\mathcal Z}f(x,y,z)\,\mathrm dx\mathrm dz \approx 1_{\{p_1\}}(y)$$ $$\int_{\mathcal X}\int_{\mathcal Y}f(x,y,z)\,\mathrm dx\mathrm dy \approx 1_{\{p_2\}}(z).$$ Specifically, I wonder whether I can conclude something like $f(x,y,z)\approx g(x)$ for some function $g$, i. e. the functional form of $f$ depends effectively only on $x$.