> [!NOTE] Theorem () > Let $g: E \subset \mathbb{R} \rightarrow \mathbb{R}$ and $a\in E$. For $i \in\{1, \ldots, n\}$, define $\pi_i: \mathbb{R}^n \rightarrow \mathbb{R}$ by > $\pi_i\left(x_1, \ldots, x_i, \ldots, x_n\right):=x_i$ and let $U_i:=\pi_i^{-1}(E):=\left\{\left(x_1, \ldots, x_n\right) \in \mathbb{R}^n: x_i \in E\right\}$. Define $f: U_i \rightarrow \mathbb{R}$ by $f\left(x_1, \ldots, x_n\right):=g\left(x_i\right)$, that is, $f(x)=g\left(\pi_i(x)\right)=g \circ \pi_i(x)$. > >If $g$ is [[Continuous maps|continuous]] at $a$, then $f$ is continuous at all points of $\pi_i^{-1}(a)=\left\{\left(x_1, \ldots, x_n\right) \in \mathbb{R}^n: x_i=a\right\}$. ###### Proof By [[Continuity of Projections of Euclidean Space]], $\pi$ is continuous on $\mathbb{R}^{\mathrm{n}}$ and therefore, by the [[Composition of Continuous Functions in Continuous|continuity of composition of continuous functions]], $f=g \circ \pi_i$ is continuous on $\pi_i^{-1}\{a\}$. # Examples ###### Example 1 We can use Proposition 3.32 and the results in section 3.5 .3 to prove the continuity of $f(x, y)=\frac{x y}{x^2+y^2}$ on $\left.\mathbb{R}^2 \backslash(0,0)\right\}$ as follows. Consider the four functions, each defined on $\mathbb{R}^2$ by $\gamma(x, y):=x, \quad \eta(x, y):=x^2, \quad \sigma(x, y):=y, \quad \tau(x, y):=y^2 .$ Proposition 3.32 tells us that the continuity of these four functions follows from the continuity (proved in First Year Analysis) of $g(t)=t$ and $h(t)=t^2$ as functions of the single real variable $t$. Now $ f(x, y)=\frac{(\gamma(x, y))(\sigma(x, y))}{(\eta(x, y))+(\tau(x, y))} $ and therefore, the continuity of $f$ on $\mathbb{R}^2 \backslash\{(0,0)\}$ follows from the continuity of the product, sum and quotient of continuous functions at points where the denominator does not vanish. ###### Example 2 A similar approach can be followed for most functions given by explicit formulas. However, the continuity of a function at points where the function is given special values (not by a formula) has to be investigated by separate arguments. The following two examples are intended to clarify what is meant by 'natural domain of definition' of a function defined by an expression involving familiar continuous functions. The natural domain of $ F(x, y)=\frac{x^2 \sin (y)}{e^x-\cosh y} $ is $\mathbb{R}^2 \backslash\{(\log (\cosh (y)), y): y \in \mathbb{R}\}$ and $F$ is continuous on this set. ###### Example 3 Similarly, $ f(x, y, z):=\left(\frac{\log (x+y)}{\sin z}, \arccos (y) \sqrt{1+\left(\cos \left(x e^z\right)\right)^2}\right) $ is continuous on $\left\{(x, y, z) \in \mathbb{R}^3: x+y>0,-1 \leqslant y \leqslant 1, z \neq n \pi, n \in \mathbf{Z}\right\}$.