Continuity in metric space

Question

Let $F: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be defined by

$$ F(x,y) = \left( x^3 y,\ \ln(x^2 + y^2 + 1),\ \cos(x - y^2) \right) $$

When trying to show why $F$ is continuous where should I start?

Answer 1

To give the answer of your question let me define first the vector valued functions.

A vector-valued function is a function from $\mathbb{R}$, or a subset of $\mathbb{R}$, to a vector space $\mathbb{R}^n$. It comprises $n$ scalar functions, one for each of the coordinates. For instance, given scalar functions $f_1$, $f_2$...$f_n$, we can construct a vector-valued function $f = (f_1, f_2, \ldots f_n) $defined by by $t \rightarrow (f_1(t), f_2(t), \ldots f_n(t))$.

Now come to your question:

The vector valued function is continuous at point $a$ if and only if all its components are continuous at $a$. Moreover, a vector valued function is continuous on the set $S$ if and only if all its components are continuous on the set S.