This is from Munkres' Analysis on manifolds:
Theorem 3.6: (a) Let $X$ be a metric space. Let $f:X\to\Bbb R^n$ have the form $f(x)=(f_1(x),\ldots,f_n(x))$. Then $f$ is continuous at $x_0$ iff each function $f_i:X\to\Bbb R$ is continuous at $x_0$.
(b) Let $f,g:X\to\Bbb R$ be continuous at $x_0$, then $f+g, f-g, f\cdot g$ continuous at $x_0$, and so is $f/g$ for $g(x_0)\ne0$.
(c) The projection function $\pi_i:\Bbb R^n\to\Bbb R$ given by $\pi(\vec x)=x_i$ is continuous.
Then author says that: "These theorems imply that functions formed from the familiar real-valued continuous functions of calculus, using algebraic operations and composites, are continuous in $\mathbb R^n$. For instance, since one knows that the functions $e^x$ and $\sin x$; are continuous in $\mathbb R$, it follows that such a function as $$f(s,t,u,v)=\frac{\sin (s+t)}{e^{uv}}$$ is continuous in $\Bbb R^4$."
I can't understand how continuity of $\frac{\sin (s+t)}{e^{uv}}$ follows from given theorems: in particular, how are $s+t$ and $uv$ continuous follow from theorem? Theorem says $f(x)+g(x)$ continuous, not something like $f(x)+g(y)$ continuous.
$f_1(s,t,u,v)=s$, $f_2(s,t,u,v)=t$,$f_3(s,t,u,v)=u$ and $f_4(s,t,u,v)=v$ are continuous functions of $(s,t,u,v)$ and so is the sine function. Apply the theorem to these functions.