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My calculus textbook has a question about proving a given statement about continuity, but it seems that my proof is much too simple for the problem. The question goes as such:

Prove that if $\lim_{Δx->0} f(c+Δx) = f(c)$ then $f$ is continuous at $c$.

first, I took $x = Δx + c$ and observed that, as Δx->0, x->c, so I rewrote the original limit $\lim_{Δx->0} f(c+Δx)$ as $\lim_{x->c} f(x)$ . We can put this in the original expression: $\lim_{x->c} f(x) = f(c)$ , the definition of continuity. Therefore, if $\lim_{Δx->0} f(c+Δx) = f(c)$ , then f is continuous at c.

This proof seems very short. Is it rigorous enough? Thanks in advance.

amsmath
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    If this is your definiton of continuity, then this is completely ok. – amsmath Aug 29 '19 at 00:37
  • I’m curious as well. If you did that step in an advanced proof, I wouldn’t think twice about it. But it almost seems to me like you’re proving that the two definitions of continuity are equivalent (or really that one implies the other) by assuming that you can perform a change of variables in the limit - which you can, but is that what you’re proving? If you wanted to be more prudent, you could use an $\epsilon$ $\delta$ proof to show that when the one limit holds, the other one does as well. – Joe Aug 29 '19 at 01:35
  • This post seems relevant: https://math.stackexchange.com/questions/167926/formal-basis-for-variable-substitution-in-limits – Joe Aug 29 '19 at 01:38

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