Q1: Is a function $f: \mathbb{Z} \rightarrow \mathbb{R}^2$ continuous?

A1: Yes, always!

A2: No, never!

A3: There are continuous functions of this form but also functions that are not continuous.

A4: The notion ‘continuous’ does not make sense for such functions.

Q2: Is the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $$ f(x_1, x_2) = x_1 \cdot x_2$$ continuous?

A1: Yes!

A2: No!

A3: One needs more information.

Q3: Is the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $$ f(x_1, x_2) = \begin{cases} 1 ~~ \text{ if } \binom{x_1}{x_2} = \binom{0}{0} \ 2 ~~ \text{ if } \binom{x_1}{x_2} \neq \binom{0}{0} \end{cases} $$ continuous?

A1: Yes!

A2: No!

A3: One needs more information.

Q4: Is the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $$ f(x_1, x_2) = \begin{cases} 0 ~~ \text{ if } \binom{x_1}{x_2} = \binom{0}{0} \ \frac{x_1 x_2}{x_1^2 + x^2_2} ~~ \text{ if } \binom{x_1}{x_2} \neq \binom{0}{0} \end{cases} $$ continuous?

A1: Yes!

A2: No!

A3: One needs more information.