Let $F:\mathbb{R}^{2} \to \mathbb{R}$, $\mathbb{I}=\mathbb{R}- \mathbb{Q}$ defined by:
$$ F(x,y) = \begin{cases} x^2+y^2 & if & x,y \in \mathbb{Q} \\ 0 & if & x \in \mathbb{I}\:or \:y \in \mathbb{I}. \end{cases} $$
$(a)$ In which points of $\mathbb{R}^2$ the function $F$ is continuous?
$(b)$ In which points of $\mathbb{R}^2$ the function $F$ is differentiable?
For (a) I know Im supposed to find 2 things(as far as I know). First, the points $(x_{0},y_{0}) \in \mathbb{R}^2$ is defined, the points where the limit of $F(x,y)$ exist as $(x,y) \to (x_{0},y_{0})$. By the definition of the function I know as every real number is rational or irrational then the function is defined for every point in $\mathbb{R}^2$. But how I determine all the points where the limit I mention before exist?
For (b) I need to know in which points of $\mathbb{R}$, the partial derivatives of $F$ exist and also are continuous, right?
Can anyone help me pointing If my idea to answer (a) and (b) are correct, and also how to end the proof? Thank so much.
Starting with part a, you already said that a function is continuous if the limit exists and is equal to the value of the function at that point. What does the sentence "the limit exists" mean? It means that if you take any sequence of $(x,y)$ going to $(x_0,y_0)$ you get the same value. Now take two different sequences, one in $\mathbb Q$, the other in $\mathbb I$. The limit of the first sequence is $x_0^2+y_0^2$, the limit of the other one is $0$. At which points do the limits coincide?
For part b, you do a similar approach.