Show that $g(y) = \int_0^1 f(x,y) dx$ is not continuous at $y =0$. 
Exercise 7.5.17 (Introduction to Real Analysis by Jiri Lebl): Define 
  $$f(x) = \left\{
  \begin{array}{lr}
    \frac{2xy}{x^4+y^2} & \text{if} & (x,y) \not= (0,0)\\
    0 & \text{if} & (x,y) =(0,0)
  \end{array}
\right.
 $$
Show that $g(y) = \int_0^1 f(x,y) dx$ is not continuous at $y =0$. Note: Feel free to use what you know about $\arctan$ from calculus, in particular that $\frac{d}{ds} [\arctan(s)] = \frac1{1+s^2}$.   

From the previous exercises, I know that $f$ is continuous at one variable, fixing another variable. In addition, I also know that $f$ is not continuous at $(0,0)$. Since the question gives us the use of the derivative of $\arctan$ as a hint, I guess that I need to use trigonometric substitution to compute the integral (?), but I am not really sure. I appreciate if you give some help. 

HINT
Let $x^2 = t$
$$\int_0^1\frac{2xy}{x^4 + y^2}dx = \int_0^1\frac{y}{(t^2 + y^2)}dt = \arctan(\frac{1}{y})$$
Can you complete the rest?
