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When proving that a continuous function $f$ can be written as the sum of a continuous odd function and a continuous even function we let $h(x)=\frac {f(x)-f(-x)}{2}$ and $g(x)=\frac {f(x)+f(-x)}{2}$. So now it suffices to show that $f(-x)$ is continuous.

$f$ continuous thus, $\forallε>0, \forallα \in D_{f},\existsδ(α,ε)>0 : \forall x \in D_{f}, |x-α|<δ \implies |f(x)-f(a)|<ε$

To prove that $f(-x)$ is also continuous do I just substitute $x$ with $-x$ and $α$ with $-α$ ? This is the part that confuses me

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    Did you find my answer at all useful? – 2017-02-08
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    @TheCount Somewhat but I'm still not convinced that is suffices to do that – 2017-02-08
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    If $f$ is continuous at $a$ then $g$ given by $g(x) =f(-x) $ is continuous at $-a$. This is nothing but writing the definition of continuity for $f$ at $a$ and changing the symbols to deal with $g$ instead of $f$. Simple plain substitution and nothing more. Most results of calculus are trivial/obvious unless they depend on completeness of real numbers. – 2017-06-13

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Note that: $f(-x)$ is just $f(x)$'s mirror image through the $y$-axis, so $f(-x)$ is continuous if and only if $f(x)$ is.

If you want to be 'official', then yes, you make the substitutions you suggested.