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Prove that the function $\displaystyle f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$. You should give a proof that is directly based on the definition of continuity.

I saw many similar questions on this website, but none of them gave me a satisfactory answer. I more or less understand how epsilon-delta proofs work but my problem lies more in the algebraic manipulations probably. So here's my "attempt".

Definition of continuity; for $f:A \to \mathbb{R}$

$$\forall \varepsilon >0 \ \exists\delta>0\ \forall x \forall x_0 \in A:|x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|<\varepsilon$$

So from what I understand, the procedure for epsilon-delta proofs are as follows:
1. Plug the point of interest $x_0$ into $|f(x)-f(x_0)|<\varepsilon$
2. Write $x$ in terms of $\varepsilon$
3. Then plug $x$ into $|x-x_0|<\delta$
4. Find $\delta$ in terms of $\varepsilon$. Done

At step 1, we have $\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|<\varepsilon$. Step 2 is where I'm stuck because of the 2nd degree polynomial in the denominator. First, I did this: $\left|\dfrac{x}{x^2+1}\right|<\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|<\varepsilon$. Then, I wasn't too sure about what to do, I thought about getting rid of the "$1$", but that would screw up my inequality. I thought about factoring $x^2+1$, but it doesn't have any real roots. I tried inverting the fraction, but that led me to nowhere. And I've already exhausted the tools I have. How do I proceed?

Apparently, epsilon-delta proofs are the most basic building blocks of analysis. And I'm already failing, this is both embarrassing and depressing...

J. W. Tanner
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Sckizel
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2 Answers2

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Hints: If $|x+1| <\delta $ where $0 <\delta <1$ then $x<-1+\delta$ and this implies $x^{2} >(-1+\delta)^{2}$ .Hence $|\frac x {x^{2}+1}+\frac 1 2|=\frac {(x+1)^{2}} {2(x^{2}+1)}<\frac {\delta^{2}} {2(1+ (-1+\delta)^{2})}$. Can you proceed?

A further hint: If you choose $\delta $ to be $<\frac 1 2$ then $(-1+\delta)< -\frac 1 2$ which implies $(-1+\delta)^{2} >\frac 1 4$.

Kavi Rama Murthy
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  • Thank you, let me think about this for a minute, I'll get you an answer. – Sckizel Feb 02 '20 at 05:20
  • Another hint maybe? – Sckizel Feb 02 '20 at 05:35
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    @Sckizel I have added one more hint. – Kavi Rama Murthy Feb 02 '20 at 05:39
  • I know I need to express $\delta$ in terms of $\varepsilon$, but I just don't see how. I give up. Thanks for trying. – Sckizel Feb 02 '20 at 06:19
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    @Sckizel Take $\delta <\sqrt {\epsilon} \sqrt {1+\frac 1 4}$. – Kavi Rama Murthy Feb 02 '20 at 06:26
  • How did you come up with $\delta$? – Sckizel Feb 02 '20 at 14:04
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    Your desired inequality is $$() \quad \left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|<\varepsilon$$ The inequality that has been proved, assuming that $|x+1|<\delta$ for some unknown value $\delta$, is $$() \quad \left|\frac x {x^{2}+1}+\frac 1 2\right|<\frac {\delta^{2}} {2(1+ (-1+\delta)^{2})}$$ Now ask yourself the question: What further inequality, in addition to $(*)$ implies $()$*? – Lee Mosher Feb 02 '20 at 15:18
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    To put this more abstractly: If you desire to prove $$() \quad a<c$$ and you know that this is true $$() \quad a<b$$ what further inequality, in addition to $()$, implies $()$? – Lee Mosher Feb 02 '20 at 15:19
  • @LeeMosher If $a<b$ and $b<c$, then that implies that $a<c$. Similarly, if $\left|\dfrac{\delta^2}{2(1+(-1+\delta)^2)}\right|<\epsilon$, then from $(*)$, I get $()$. Now, how do you justify $\left|\frac{\delta^2}{2(1+(-1+\delta)^2)}\right|<\epsilon$? Is it an arbitrary choice made to fit the problem? – Sckizel Feb 02 '20 at 21:40
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    Your goal is to find a value of $\delta$. By setting up that inequality and solving for $\delta$, you've found it. – Lee Mosher Feb 02 '20 at 21:47
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    Remember, to prove $\delta$ exists, you must find the appropriate value. For this problem, that comes down to solving that inequality. – Lee Mosher Feb 02 '20 at 21:47
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$f(x)$ is continuous at $f(-1)=-1/2$ is finite and euals both the right and the left limits: $$\lim_{x\rightarrow -1^+} f(x)= \lim_{h \rightarrow 0} \frac{-1+h}{1+(-1+h)^2}=-\frac{1}{2}$$ $$\lim_{x\rightarrow -1^-} f(x)= \lim_{h \rightarrow 0} \frac{-1-h}{1+(-1-h)^2}=-\frac{1}{2}$$

Z Ahmed
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  • Thank you for your answer, but this is not what I need. The problem asks explicitly to use the epsilon-delta definition of continuity. – Sckizel Feb 02 '20 at 05:21