$f: X \to Y, g: X \to Z$. We define their direct sum
$$f \oplus g : X \to Y \times Z $$
as
$$f \oplus g (x) := (f(x), g(x))$$
$f: X \to \R, g: X \to \R.$ We give $\R^2$ the Euclidean metric.
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If $x_0 \in X$, then $f$ and $g$ are both continuous at $x_0$ if and only if $f \oplus g$ is continuous at $x_0$.
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$f$ and $g$ are both continuous if and only if $f \oplus g$ is continuous.
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The addition function $(x, y) \mapsto x + y$
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the subtraction function $(x, y) \mapsto x - y$
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the multiplication function $(x, y) \mapsto xy$
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the maximum function $(x, y) \mapsto \max{ x, y }$
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the minimum function $(x, y) \mapsto \min{ x, y }$
are all continuous functions from $\R^2$ to $\R$.
The division function $(x, y) \mapsto x / y $ is a continuous function from $\R \times \R \setminus \set{ 0 }$ to $\R$.
$\forall c \in R$, the function $x \mapsto cx$ is a continuous function from $\R$ to $\R$.
Let $(X, d) $ be a metric space, let $f: X \to \R $ and $g: X \to \R $ be functions. Let $c $ be a real number.
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If $x_0 \in X $ and $f $ and $g $ are continuous at $x_0 $, then the functions $f+g: X \to \R, f-g: X \to \R, fg: X \to \R, \max{ f,g }: X \to \R, \min{ f,g }: X \to \R, $ and $cf : X \to \R $ are also continuous at $x_0$. If $\forall x \in X, g(x) \neq 0$, then $f/g: X \to \R $ is also continuous at $x_0 $.
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If $f$ and $g$ are continuous, then the functions $f + g : X \to \R $, $f -g : X \to \R $, $fg: X \to \R$, $\max{ f,g }: X \to \R $, $\min{ f,g }: X \to \R $, and $cf : X \to \R $ are also continuous at $x_0$. If $\forall x \in X, g(x) \neq 0$, then $f/g : X \to \R $ is also continuous at $x_0$.