I would like to prove a chain rule for limits (from which the continuity of the composition of continuous functions will clearly follow): if [tex]\lim_{x\to c} \, g(x)=M[/tex] and [tex]\lim_{x\to M} \, f(x)=L[/tex], then [tex]\lim_{x\to c} \, f(g(x))=L[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

Can someone please tell me if the following proof is correct? I am a complete newbie to writing proofs, so I might have made several basic mistakes.

The second postulate means that there exists a [tex]\delta _1[/tex] for which the following is true for all [tex]x[/tex] in the domain:

[tex]0<|x-M|<\delta _1\Rightarrow |f(x)-L|<\epsilon[/tex]

By substituting [tex]x[/tex] with [tex]g(x)[/tex] we get the following which is true for all [tex]g(x)[/tex] in the domain:

[tex]0<|g(x)-M|<\delta _1\Rightarrow |f(g(x))-L|<\epsilon[/tex]

The first postulate means that there exists a [tex]\delta[/tex] for which the following is true for all [tex]x[/tex] in the domain:

[tex]0<|x-c|<\delta \Rightarrow |g(x)-M|<\delta _1[/tex]

Thus, by transitivity:

[tex]0<|x-c|<\delta \Rightarrow |f(g(x))-L|<\epsilon[/tex]

QED

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# Chain rule for limits

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