Proving f is continuous using epsilon and delta

Question

I need to prove that f is continuous in (1,0):

$$\frac{(x-1)y^2}{(x-1)^2+y^2}-x \; for \;x \ne (1,0) \; and \; f(1,0) = -1$$

I came up with the following until now:

$$\left|\frac{(x-1)y^2}{(x-1)^2+y^2}-x+1\right|\leq \left|\frac{(x-1)y^2}{(x-1)^2+y^2}\right|+\left|(1-x) \right|$$

However, I do not know how to continue.

Thanks in advance for your help.

Answer 1

hint

use the fact that

$$\Bigl|\frac{(x-1)y^2}{(x-1)^2+y^2}-x+1\Bigr|\le \frac{|x-1|y^2}{y^2}+|x-1|$$

or $$|f(x,y)-f(1,0)|\le 2|x-1|$$