Are random variables independent if their joint density function doesn't impose a constraint?

Question

Say I have the following joint density function:

$$f_{X,Y} = \begin{cases} c(x + y^2) & 0 \leq x \leq 1, 0 \leq y \leq 1 \\ 0 & \text{ otherwise }\end{cases}$$

To me it looks like $f_{X,Y}$ is a function of two "independent variables" (in the sense we mean when discussing parametric equations) -- a given value of $x$ does not seem to impose any particular value of $y$ -- does this necessarily always imply $X$ and $Y$ are independent random variables (in the sense that $P(X \in A, Y \in B) = P(X \in A)P(Y \in B)$)?

Answer 1

No. For $X$ and $Y$ to be independent, you must have $f(x,y)=f_{X}(x)f_{Y}(y)$ for all $x$ and $y$.