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I have a question related to another post I made but I thought it’d be best to create a new thread.

I was trying to understand the definition below and I noticed that it was an if statement. I interpreted it as meaning that if the limit of f(x,y) as it approaches (a,b) = f(a,b) then it is continuous as at (a,b).

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However, I noticed that when we solve limits, we consider whether or not they are continuous and then if they are, we plug in the values. This would mean that we are using the reverse of the definition, i.e., if it’s continuous at (a,b) then the limit of f(x,y) as it approaches (a,b) = f(a,b).

Since it is an if statement, is the reverse necessarily true? Am I missing something?

Oofy2000
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    The duplicate answers your question: in definitions, "if" means "iff". Whatever the context and the definition, in practical applications both directions are useful (sometimes one, sometimes the other one). – Anne Bauval Apr 12 '23 at 09:23
  • Perfect. Thanks guys. I didn’t realise it had already been asked. – Oofy2000 Apr 12 '23 at 09:30

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First of all, all definitions are an "if and only if" statement, but the "only if" is sort of assumed by default, since it means that we use the label (being continuous at a point $p$) only when a certain condition (limit is equal to the value at $p$), so we're saying that the word has no other meaning, which is taken for granted.

Second of all, when you have a limit $$\lim_{x\rightarrow x_0}F(x)$$of some expression in $x$, one way to solve it is to prove $F$ is continuous at $x_0$. For example, if $F(x)=f(x)+g(x)$ and both $f,g$ are continuous at $x_0$, then so is $F$. In general, $F$ may be a combination (product, sum, exponential, etc.) of other functions which are continuous at $x_0$, so $F$ is continuous as well (provided $x_0$ is in the domain of $F$, for example, $x$ is continuous at $0$, but $\frac{1}{x}$ is not defined at $0$, least of all continuous). If you're in that situation, you can calculate the limit $\lim_{x\rightarrow x_0}F(x)$ by simply plugging $x_0$ to get $F(x_0)$. If $F$ is not continuous at $x_0$ or not defined at $x_0$ at all, you need another strategy.

Alessandro
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