# Continuity of a Function .... Conway, Definition 1.7.1 .... ....

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In summary, the conversation discusses the definition of continuity in John B. Conway's book "A First Course in Analysis". The group focuses on Definition 1.7.1 in Section 1.7, and there is confusion about the notation used and the clarity of the definition. It is mentioned that there is a typo in the definition, and two different ways of defining continuity are discussed.
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I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 1: The Real Numbers ... and in particular I am focused on Section 1.7: Continuous Functions ...

I need help with clarifying Definition 1.7.1 ...Definition 1.7.1 reads as follows:
View attachment 9497My question is as follows:

Is the above definition clear and correct? Is it usual?It seems to me Conway has defined continuity at any point $$\displaystyle x \in X$$ ... so why bother mentioning $$\displaystyle a$$ ... ?Can someone please clarify Conway's approach to continuity ...

Peter

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Peter said:
I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 1: The Real Numbers ... and in particular I am focused on Section 1.7: Continuous Functions ...

I need help with clarifying Definition 1.7.1 ...Definition 1.7.1 reads as follows:
My question is as follows:

Is the above definition clear and correct? Is it usual?It seems to me Conway has defined continuity at any point $$\displaystyle x \in X$$ ... so why bother mentioning $$\displaystyle a$$ ... ?Can someone please clarify Conway's approach to continuity ...

Peter
And HOW did he define "at any point $$\displaystyle x\in X$$? As being continuous at every $$a\in X$$. That wouldn't make sense unless he had defined "continuous at a" first!

HallsofIvy said:
And HOW did he define "at any point $$\displaystyle x\in X$$? As being continuous at every $$a\in X$$. That wouldn't make sense unless he had defined "continuous at a" first!
Hi HallsofIvy ...

This is Conway's first and only definition of continuity for a real valued function of a real variable ...

I feel that there is a typo in it since as it stands it doesn't seem to make sense ...

Peter

Peter said:
This is Conway's first and only definition of continuity for a real valued function of a real variable ...

I feel that there is a typo in it since as it stands it doesn't seem to make sense ...

Peter
You are quite correct, and once again you have found a typo in a respected textbook. Here, Conway claims to be defining continuity at $a$, but he actually defines continuity at $x$. In other words, he changes notation halfway through the definition.

Peter said:
Is the above definition clear and correct? Is it usual?
There are two ways of defining continuity. One is the $\varepsilon$-$\delta$ definition, the other (used here by Conway) is to use sequences. Both definitions are "usual", and they are equivalent to each other. Each of them is useful in different contexts. Whichever of them is taken as the initial definition, most authors find it helpful to introduce the other one later, and to prove their equivalence.

## 1. What is the definition of continuity of a function?

The definition of continuity of a function, according to Conway's Definition 1.7.1, is that a function f is continuous at a point a if the limit of f(x) as x approaches a exists and is equal to f(a).

## 2. How is continuity of a function different from differentiability?

Continuity of a function refers to the smoothness of the function at a specific point, while differentiability refers to the smoothness of the function over an interval. A function can be continuous but not differentiable, but if a function is differentiable, it must also be continuous.

## 3. What are the three types of continuity?

The three types of continuity are pointwise continuity, uniform continuity, and local continuity. Pointwise continuity means that the function is continuous at each individual point. Uniform continuity means that the function is continuous over an entire interval. Local continuity means that the function is continuous at a specific point and its nearby points.

## 4. How can you determine if a function is continuous?

A function is continuous if it satisfies the definition of continuity, which means that the limit of the function at a point exists and is equal to the value of the function at that point. This can be determined by evaluating the function at the point and taking the limit as x approaches that point.

## 5. What are some real-world applications of continuity of a function?

Continuity of a function is important in many areas of science and engineering. It is used in physics to describe the smoothness of motion, in economics to model continuous changes in variables, and in computer science to create smooth graphics and animations. It is also crucial in calculus and other areas of mathematics to prove theorems and solve problems.

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