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Math Help - Proof that the range of f is all of R

  1. #1
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    Proof that the range of f is all of R

    Hey all, I need some help with this proof for my analysis class. I have a proof that I wrote up but I'm not sure if it's correct. I'll write the claim and my idea of a proof below and if anyone could tell me what's wrong with my logic or if there's a better way to write the proof, I'd really appreciate it. I'm just not sure how to go about this so what I'm providing is my best guess.

    Claim: Assume f is a continuous real-valued function with domain (-\infty , \infty ). Assume \lim_{x \rightarrow \infty} f(x) = \infty and \lim_{x \rightarrow - \infty} f(x) = - \infty. Prove that the range of f is (-\infty , \infty ).

    Proof: Assume the hypotheses. Since f is continuous at every point in its domain, we have that \lim_{x \rightarrow x_0} f(x) = f(x_0) for all x_0 \in (-\infty , \infty ). Then by the definition of limit we have that for all \varepsilon > 0, there exists a \delta > 0 such that |f(x) - f(x_0)| < \varepsilon whenever |x - x_0| < \delta. That is, f(x_0) - \varepsilon < f(x) < f(x_0) + \varepsilon. So if we let x_0 \rightarrow \infty, then f(x_0) \rightarrow \infty and so f(x) < \infty. Also, if we let x_0 \rightarrow -\infty, then f(x_0) \rightarrow -\infty and so -\infty < f(x). Then -\infty < f(x) < \infty, so the range of f is (-\infty , \infty ).

    Again, I know it seems sketchy, so please help me identify what I need to say to make this more precise.
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  2. #2
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    Well, how about this:

    Take n\in \mathbb{N} and N_1,N_2 such that if x<N_1 then f(x)\leq -n and if x>N_2 then f(x)\geq n (these N's exist by your limit conditions) then by the Intermediate value theorem we have that for every y\in (-n,n) there exists a z\in \mathbb{R} such that f(z)=y. Since this is true for any n we get the result.
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  3. #3
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    Quote Originally Posted by Jose27 View Post
    Well, how about this:

    Take n\in \mathbb{N} and N_1,N_2 such that if x<N_1 then f(x)\leq -n and if x>N_2 then f(x)\geq n (these N's exist by your limit conditions) then by the Intermediate value theorem we have that for every y\in (-n,n) there exists a z\in \mathbb{R} such that f(z)=y. Since this is true for any n we get the result.
    Wow, that works very nicely. And it makes a lot more sense that I would need to use the IVT, since we just learned about it not too long ago. Thanks for your help!
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