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I always thought that this theorem was true, but today when asking about the proof of the chain rule in calculus I realized that it is false. I understand the counterexample, but now I don't understand where my proof fails. (Here is my attempt to prove the chain rule in calculus and the counterexample is in the answer).

Theorem:

Let $A,B\subseteq \mathbb{R}$, $f:A\longrightarrow B$, and $g:B\longrightarrow \mathbb{R}$ functions. Let $a\in R$ be an accumulation point of $A$. Let's suppose that $\lim_{x\to a}f(x)=b$ and $\lim_{x\to b} g(x)=l$. Then $\lim_{x\to a}(g\circ f)(x)$ exists and equals $l$.

Proof:

Let $\epsilon>0$ be an arbitrary number in $\mathbb{R}$. Since $\lim_{x\to b} g(x)=l$ then there is $\delta >0$ in $\mathbb{R}$ such that $\forall x\in B (0<|x-b|<\delta \longrightarrow |g(x)-l|<\epsilon)$. Since $\lim_{x\to a}f(x)=b$ then there is $\delta '>0$ such that $\forall x\in A(0<|x-a|<\delta '\longrightarrow |f(x)-b|<\delta)$. But then $\forall x\in A(0<|x-a|<\delta '\longrightarrow |g(f(x))-l|<\epsilon)$. This means that $\lim_{x\to a}(g\circ f)(x)$ exists and equals $l$.

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The issue here is that you have no guarantee that $g$ is continuous.

The problem in the proof itself is that while it may be true that $\forall x \in A(0 < |x - a| < \delta' \rightarrow |f(x) - b| < \delta)$, it is not necessarily true that $\forall x \in A(0 < |x - a| < \delta' \rightarrow 0 < |f(x) - b| < \delta)$, and if $f(x) = b$ so that $|f(x) - b| = 0$, then this cannot then be used as an input to $g$.

However, if $g$ is continuous at $b$, then it is true that $g(b) = l$ (by definition of continuity), and so $|g(b) - l| = 0 < \epsilon$ for all $\epsilon > 0$.

Thus, it is no longer needed that $0 < |x - b| < \delta$ for $|g(b) - l| < \epsilon$, and instead, it will suffice to require that $|x - b| < \delta$.

This then allows the remainder of your proof to work.

Edit: An alternate sufficient condition would be to require that $\exists \delta_1 > 0$ such that $\forall x \in A(0 < |x - a| < \delta_1 \rightarrow 0 < |f(x) - b|$.

Then, simply take $\min(\delta_1, \delta')$ to be the new value of $\delta'$ used in the proof.

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  • $\begingroup$ This a very important technical detail which I am sure many book authors have missed. Thanks for clarifying it here. $\endgroup$ – Paramanand Singh Nov 17 '13 at 5:03
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Suppose that $g$ is not continuous at $b$, so that $l \neq g(b)$, and suppose that for any choice of $\varepsilon$, is some $x^\ast_\varepsilon$ satisfying $0 < |x^\ast_\varepsilon - a| < \delta'(\varepsilon)$ and $f(x^\ast_\varepsilon) = b$ (this can happen, for example, if $f$ is the constant function $f(x) \equiv b$). If both of these things happen, then $$|g(f(x^\ast_\varepsilon)) - l| = |g(b) - l| > 0.$$ It then follows that if we choose $\varepsilon < |g(b) - l|$, we have $$|g(f(x^\ast_\varepsilon)) - l| = |g(b) - l| > \varepsilon \text{ even though } |x^\ast_\varepsilon - a| < \delta'(\varepsilon).$$ Hence your choice of $\delta'(\varepsilon)$ doesn't work in this scenario.

If you assume that either $g$ is continuous or $f(x) \neq b$ for $x$ close to $a$, then your proof will work.

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