# Limits of functions .... D&K Lemma 1.3.3 .... another question ....

• MHB
• Math Amateur
Name]"In summary, the conversation discusses Chapter 1: Continuity and the proof of Lemma 1.3.3 in the book "Multidimensional Real Analysis I: Differentiation" by Duistermaat and Kolk. The authors state that the necessity of the proof is obvious, but the reader is confused and needs help understanding it. The proof uses the epsilon-delta definition of continuity to show that for a function to be continuous at a point, the limit of the function must exist at that point and be equal to the value of the function at that point. The expert provides a rigorous proof to clarify the "obvious" statement in the proof.
Math Amateur
Gold Member
MHB
I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of the proof of Lemma 1.3.3 ...

Duistermaat and Kolk"s proof of Lemma 1.3.3. reads as follows:View attachment 7681In the above proof we read:

" ... ... The necessity is obvious ... ... "

Presumably this means that if $$\displaystyle \lim_{x \rightarrow a} f(x) = b$$ then for every sequence $$\displaystyle (x_k)_{ k \in \mathbb{N} }$$ with $$\displaystyle \lim_{k \rightarrow \infty } x_k = a$$ we have $$\displaystyle \lim_{k \rightarrow \infty } f( x_k ) = b$$ ... ...Although D&K reckon that it is obvious I cannot see how to (rigorously) prove the above statement ...

Can someone please demonstrate a rigorous proof ...
Help will be much appreciated ... ...

Peter

Hi Peter,

The point is that limits are not defined in exactly the same way for functions and sequences.

$\lim_{x\to a}f(x)=b$ means that for any $\varepsilon>0$ we can find $\delta>0$ such that $|x-a|<\delta\Rightarrow|f(x)-b|<\varepsilon$.

$\lim_{k\to\infty}x_k=a$ means that, for every $\delta>0$, there exists a $k_0\in\mathbb{N}$ such that $k>k_0\Rightarrow|x_k-a|<\delta$.

Now, for $k>k_0$, we have $|x_k-a|<\delta$, and, because of the hypothesis, $|f(x_k)-b|<\varepsilon$; this is precisely what $\lim_{k\to\infty} f(x_k) = b$ means.

Hi Peter,

I can understand your confusion with this statement. "Obvious" is a subjective term and what may seem obvious to the authors may not be obvious to all readers. In this case, they are referring to the fact that for a function to be continuous at a point a, the limit of the function must exist at that point and be equal to the value of the function at that point.

To prove this, we can use the epsilon-delta definition of continuity. Let \epsilon > 0 be given. Since \lim_{x \rightarrow a} f(x) = b, we know that there exists a \delta > 0 such that for all x \in \mathbb{R}, if 0 < |x - a| < \delta, then |f(x) - b| < \epsilon.

Now, let (x_k)_{k \in \mathbb{N}} be a sequence with \lim_{k \rightarrow \infty} x_k = a. This means that for any \delta > 0, there exists a natural number N such that for all k > N, we have |x_k - a| < \delta. Therefore, for all k > N, we have |f(x_k) - b| < \epsilon, which implies that \lim_{k \rightarrow \infty} f(x_k) = b.

Thus, we have shown that for any sequence (x_k)_{k \in \mathbb{N}} with \lim_{k \rightarrow \infty} x_k = a, we have \lim_{k \rightarrow \infty} f(x_k) = b. This completes the proof of Lemma 1.3.3.

I hope this helps clarify the "obvious" statement in the proof. Let me know if you have any other doubts.

## 1. What are the limits of functions?

The limits of functions refer to the values that a function approaches as its input approaches a certain value. It can also be thought of as the output of a function at a particular point.

## 2. What is D&K Lemma 1.3.3?

D&K Lemma 1.3.3, also known as the "Squeeze Theorem", is a theorem in calculus that states if two functions have the same limit at a certain point, and a third function is squeezed between them, then the third function also has the same limit at that point.

## 3. How is D&K Lemma 1.3.3 useful?

D&K Lemma 1.3.3 is useful in proving the limits of complex functions by simplifying them into simpler functions that have known limits. It is also used in proving theorems in calculus, such as the Intermediate Value Theorem.

## 4. Can D&K Lemma 1.3.3 be applied to all functions?

No, D&K Lemma 1.3.3 can only be applied to functions that are continuous at a certain point. This means that the function has no breaks or jumps at that point.

## 5. Are there other important lemmas in calculus besides D&K Lemma 1.3.3?

Yes, there are many important lemmas in calculus, such as the Mean Value Theorem, Rolle's Theorem, and Intermediate Value Theorem. These lemmas are used to prove various theorems and properties in calculus.

Replies
2
Views
1K
Replies
2
Views
1K
Replies
2
Views
1K
Replies
4
Views
1K
Replies
6
Views
1K
Replies
2
Views
1K
Replies
2
Views
1K
Replies
2
Views
941
Replies
2
Views
1K
Replies
2
Views
1K