Let $f(x,y)$ be a real-valued function on the unit square $[0,1]^2$. Suppose that $f(x,y)$ is Riemann integrable along each straight line. Does this imply that $f$ Riemann integrable on the square? Does the answer change if we only suppose that $f$ is integrable along all vertical lines $x=c$ and integrable along all horizontal lines $y=c$? 



If a real-valued bivariate function on the unit square is integrable along each line, is it integrable on the square?
No, consider the function 
$$g(x,y) = \cases {0,&if $(x,y)=(0,0)$\\
xy^2/(x^2+y^6) &otherwise}$$
taken from Exercise 4.7 of Baby Rudin (3ed) via [this Math.SE post](https://math.stackexchange.com/questions/174816/discontinuous-functions-that-are-continuous-on-every-line-in-bf-r2). Along every line it is continuous and hence Riemann integrable, but it is unbounded along the curve $x=y^3$, and an unbounded function cannot be Riemann integrable.


