Is there an example of a function $f:[0,1] \times [0,1] \to \mathbb{R}$ such that for all $x \in [0,1]$ the function $\phi(y) = f(x,y)$ is continuous in $y$ and for all $y \in [0,1]$ the function $\psi(x) = f(x,y)$ is continuous in $x$ but the function defined by $$J(x) = \int_0^1{f(x,y)}dy$$ is discontinuous at some $x \in [0,1]$?
I'm asking this because a theorem about the continuity of parametric integrals requires the function $f$ to be continuous in both variables at the same time (continuous as a function from $X \to \mathbb{R}$ with $X = [0,1]\times [0,1]$, but I can't find a counterexample for the case mentioned above.
EDIT: I know the difference between a function being continuous in two variables separately and jointly. What I want to know if there is a function that is continuous in two variables separately and the integral above is discontinuous. The function $$f(x,y) = \frac{2xy}{x^2+y^2}$$ is an example for a function that is not jointly continuous. But the parametric integral of this function IS still continuous.
