Reciprocal Function is Continuous on Real Numbers without Zero
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Theorem
Let $\R_{\ne 0}$ denote the real numbers excluding $0$:
- $\R_{\ne 0} := \R \setminus \set 0$.
Let $f: \R_{\ne 0} \to \R$ denote the reciprocal function:
- $\forall x \in \R_{\ne 0}: \map f x = \dfrac 1 x$
Then $f$ is continuous on all real intervals which do not include $0$.
Proof
From Identity Mapping is Continuous, the real function $g$ defined as:
- $\forall x \in \R: \map g x = x$
is continuous on $\R$.
From Constant Mapping is Continuous, the real function $h$ defined as:
- $\forall x \in \R: \map x h = 1$
We note that $\map g 0 = 0$.
The result then follows from Quotient Rule for Continuous Real Functions:
- $\map f x = \dfrac {\map h x} {\map g x}$
is continuous wherever $\map g x \ne 0$.
$\blacksquare$