# Questions tagged [epsilon-delta]

For questions regarding $\varepsilon$-$\delta$ definitions of limits, continuity of functions and $\varepsilon-N$ definition of limit of sequences.

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### What is range of values that the word 'nearby' supposed to represent in this informal definition of continuity.

In my book it gave two informal explanation for the concept of continuity. I had doubt in the second explanation but I cleared it by asking it here. The explanation is , Suppose a function f has the ...
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### How can I prove $\lim_{x \to a} f(x)=L$ implies $\lim_{x \to 3a}f(x/3)=L$ with the epsilon-delta definition of the limit?

How can I prove $\lim_{x \to a} f(x)=L$ implies $\lim_{x \to 3a}f(x/3)=L$ with the epsilon-delta definition of the limit? I have tried doing this by writing down separately the epsilon-delta ...
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### Using the epsilon-delta definition of continuity to prove a linear function is continuous at c

I'm familiar with using the definition to prove a polynomial is continuous at point c, but I've yet to use it with respect to linear functions. Example: Prove $h(x)$ is continuous at 4. $$h(x)=3x-1$$ ...
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### Clarification about Proof of the Product of two Continuous Functions being Continuous

Almost all $epsilon$-$delta$ proofs of the theorem below either place extra conditions on $\epsilon$, like on page 77 of Tom M Apostol's "Mathematical Analysis," or are copies of the top ...
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### Is my attempt at a proof valid?

Warning: I have no formal education in these, so sorry if my attempt is horrible. I've recently learnt the concept behind a limit, i.e. the epsilon-delta definition and it seems really cool. I've ...
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### Continuity of a pointwise maximum function of probability distributions

Consider a pair of finite alphabets $\mathcal{X}$ and $\mathcal{Y}$. Let $P_{Y|X}$ be a conditional probability distribution and let $Q_Y$ be a full-rank probability distribution. I am looking at the ...
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### Proof Verification for $e^x$ is not uniformly continuous on R

I would like some feedback on my attempt / whether or not this is a valid approach as it differs from the other solutions I have been able to find. I am aware that similar questions have been asked ...
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### Using $\epsilon$-$\delta$ definition of limit to prove a limit doesn't exist.

How can I use the $\epsilon$-$\delta$ definition of limit to prove that the following limit doesn't exist? $$\lim_{x\to1} \sin(\frac{1}{x-1})$$ So far, I have tried to write out the definition of ...
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### Let $g$ be continuous on an interval $A$ and let $F$ be the set of points where $g$ fails to be one-to-one. Show $F$ is either empty or uncountable.

Intermediate Value Theorem (I'll be using this): Let $f : [a,b] \rightarrow \mathbb{R}$ be continuous. If $L$ is a real number satisfying $f(a) < L < f(b)$ or $f(a) > L > f(b)$, then there ...
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I need to prove that $$\lim_{(x,y)\to(0,0)} \frac{xy(x^2-y^2)}{x+y} = 0.$$ I checked by approaching origin from all directions by substituting $$y = mx$$ and doing $$\lim_{(x)\to(0)} \frac{xy(x^2-y^... 1 vote 3 answers 74 views ### Prove that \lim_{x \to 0} x \sin\left(\frac1x\right)=0 Hi everyone im struggling a bit with this question. I kinda get the idea of epsilon-delta proof, and have seen you can choose \epsilon = \delta I don't get why tho. Could someone explain that to me? ... 1 vote 0 answers 100 views ### Simplifying asymptotics. Question: If for every \epsilon>0 however small such that \epsilon\leq r\leq 1-\epsilon, then simplify the asymptotic$$f(r)=\mathcal{O}\left(\frac{r}{1-r^2} \ \log\left( \frac{r}{1-r^2} \... 41 views

### Proving Riemann Integrability if only if there exists $L \in \mathbb{R}$ (Bartle's exercise 7.1 no.3)

Question : Show that $f : [a,b]\to \mathbb R$ is Riemann integrable on $[a,b]$ if and only if there exists $L\in \mathbb R$ such that for every $\epsilon \gt 0$ there exists $\delta_\epsilon \gt 0$ ...