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I'm self studying real analysis and currently reading about the limits of functions. Naturally everything in the chapter is about determining if a limit exists at a single point. But what about showing that a given function has limits over its entire domain? Take the class of non-rational polynomial functions. Obviously these have a limit at every $x_0$ in $\mathbb R$. My questions is, can it be proven that a given polynomial (or even the class of all non-rational polynomial functions) has a limit at every point using just what I've learned so far (formal definition of the limit of a function, the relationship between the limit of a sequence and a function, algebra of limits etc)? Or is this something that would require more knowledge?

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    Can you give an example of what you mean by a non-rational polynomial function and what you mean by a limit at a single point?2011-10-26
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    I think you're talking about continuity here, which should be the next chapter in the book..2011-10-27
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    The theorems relating sums and products to limits immediately imply that the set $S$ of all functions $f: \mathbb{R} \to \mathbb{R}$ with the property that $\lim_{x \to c} f(x)$ exists for all $c \in \mathbb{R}$ is closed under the usual multiplication and addition of functions. So once you have proved (by hand) that constant functions are in $S$, and the function $g$ given by $g(x) = x$ is in $S$, it follows that any polynomial is in $S$. To be formal about this you might need a short proof by induction, but you don't need "more knowledge." Maybe someone will flesh this out in an answer.2011-10-27
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    +1 for thinking ahead of the text. @lhf is right; you're thinking about _continuity_, which is vastly important and should be next.2011-10-27
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    No, I'm thinking about proving that a given function (or possibly a class of functions) has a limit at each point in it's domain. It's been about 10 years since I had Calculus, but continuity might possibly be used to prove it but I'm not sure. Can the result be generalized from a single point to an entire domain. Perhaps I shouldn't have used the term 'non-rational'. What I mean is a polynomial that if not of the form p(x)/q(x).2011-10-27
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    @CritChamps. So, what you call "non-rational polynomial function" is just a polynomial function. And a function like $p(x)/q(x)$, with $p(x), q(x)$ polynomials is a rational function.2011-10-27
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    Is the question why the book focuses on proving things about limits at a single point, when in applications one often needs the existence of limits on large sets of points? The "point" (ha, ha) is that "for all $c \in D$, $\lim_{x \to c} f(x)$ exists" is a "for all" statement defined in terms of limits at a point. To prove it, you can fix a symbol $c$, assume $c \in D$, and prove that $\lim_{x \to c} f(x)$ exists. The existence of limits for large sets of $c$ is definable in terms of symbols, what is going on at a single point, and a "for all" quantifier. No new concepts are needed.2011-10-27
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    @lhf. I think that CritChamp is right: he's not necessarily talking about continuity. For instance, a function like $f(x) = 0$ for every $x\neq 0$ and $f(0)=1$ is one of "his" functions because it has a limit for every point. But it's not a continuous function.2011-10-27

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