# 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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### Epsilon Delta Proof of the Derivative of X Cubed

Right, so this was a question I randomly pondered up while I was watching through a BlackPenRedPen video, in which he shows off a question similar to the one I'm about to ask, except with f(x) = $x^2$ ...
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### Where was the formal definition of a limit first published?

I've read that Bolzano was one of the first to formalize a limit with the epsilon-delta definition, but I can't find a text of his that shows his definition. Does anyone know what publication the ...
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### In epsilon-delta proofs, is delta found be equal to a value or less than or equal to a value?

In epsilon-delta proofs, when specifying delta, is it correct to specify the delta is equal to a value or delta is less than or equal to a value? Which statement if more correct and why? Given epsilon ...
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### Supremum of all deltas for a given epsilon

A function $f:\mathbb{R} \to\mathbb{R}$ is continuous at a point $c$ if, for every $\varepsilon >0$ there exists a $\delta > 0$ such that $|x-c| < \delta$ implies $|f(x)-f(c)|<\varepsilon$....
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### Prove that a family of functions bounded by a $L^1$ function is uniformly integrable: $\varepsilon$-$\delta$ proof

Let $\mu$ be a finite measure and let $(f_i)_{i\in I}$ be a sequence of functions such that $$f_i\le (k g)^{1/2} \quad \forall i\in I,$$ where $k>0$ is a constant and $g\in L^1(\mu)$. I am trying ...
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### Proving every continuous function on closed interval is uniformly continuous by smallest delta

When proving the statement above, I completely understand the textbook proof with convergent subsequences and contradiction, but I wonder: when function is continuous by classic epsilon-delta ...
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### What does $V_{\delta_x} (x)$ mean?

I was solving exercises on real analysis by Introduction to Real Analysis by Robert Bartle and Donald Sherber forth edition exercise 5.3 page 141 and I am confused what does $V_{\delta_x}(x)$ in Let ...
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### Continuity at a point doesn’t imply continuity at some small interval around it

I wanted to prove that point continuity does not imply local continuity. I have already seen examples like the function $f(x) = x^{2}$ if $x$ is rational and 0 if irrational, which are nowhere ...
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### proving $\lim_{x\to \infty}$ ${x+3\over x-2}$=1 using delta epsilon method

I need to prove $\lim_{x\to \infty}$ ${x+3\over x-2}$=1 using delta-epsilon method. Here's the part I tried but I'm stuck in the middle in choosing M with $\varepsilon$. Let $\varepsilon$ >0 be ...
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### Why is the limit of a function defined at only a cluster point of the domain?

My textbook gives the following definition of a limit: Let $A\subseteq\mathbb R$, and let $c$ be a cluster point of $A$. For a function $f:A\to\mathbb R$, a real number $L$ is said to be the limit of ...
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### Nature of the set of deltas in limit definition

I was proving the fact that if f is differentiable at a, then (cf)’(a)=cf(a)’ After showing it, I started to play with some ideas and wondered from intuition how to see that depending on the value of ...
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### Prove that $\lim_{n\to\infty}a_n=1/2$ by definition

Suppose that $$a_n=\frac{n^2-2n+1}{2n^2+4n-1}.$$ I am trying to show that for each positive number $\epsilon$, there is a number $N$ such that $|a_n-L|<\epsilon$ whenever $n>N$. Attempt ...
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### Algebraic rules of absolute value

In evaluating elementary $\epsilon$, $\delta$ proofs of limits, one often sees the following sort of move: $$\left|2x - 8\right| = \left|2(x-4)\right| = 2\left|x - 4\right| \dots$$ (See e.g. here (14:...
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### Continuity of $f(n):=n,6n\in\mathbb{Z}$

Suppose $f:\frac{1}{6}\mathbb{Z}\to\frac{1}{6}\mathbb{Z}$ is a function defined by $f(n):=n,6n\in\mathbb{Z}$. Is $f$ continuous at $x=0$? A graph of the function somewhat points towards this answer: ...
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### Proof of limit by epsilon-delta method: $\displaystyle\lim_{x \to 9} \sqrt{x-5} = 2$

The question is: Prove the below limit statement: $\displaystyle\lim_{x \to 9} \sqrt{x-5} = 2$ From my understanding of the textbook (Thomas' Calculus), the proof is done in 3 steps: Write both the ...
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### Epsilon delta is super redundant, isn't it? [closed]

I have a very different question. I was looking into $\epsilon-\delta$ definition of limit. I actually understand the idea, but what I'm wondering is why it was necessary to come up with such an idea ...
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### Solving quadratic inequalities which cannot be factorized

This question is a step involved in solving the below question: Find an open interval about c on which the inequality $|f(x)-L|<\epsilon$ holds. $f(x) = 4+x-3x^2$, $L=2$, $c=1$, $\epsilon=0.01$ I ...
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### Example of a limit where the Epsilon delta definition wins

In my high school Limit of a function was introduced as below. We say $\lim_{x\to a^-} f(x)$ is the expected value of $f$ at $x = a$ given the values of $f$ near $x$ to the left of $a$. This value is ...
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### Finite decimal continuity

I am learning multi-var calculus using the textbook "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach". Here is a definition that's puzzling me. Let $\mathbb{D}$ ...
### can we say a function $f:[a,b]\to \Bbb{R}$ is continuous at the end points?
$f:[a,b]\to \Bbb{R}$ is said be continuous at a point $m$ that means whenever a sequence $x_n \in[a,b]$ converge to ${m}$ the image ${f(x_n)}$ converge to ${f(m)}$ and by applying this definition of ...