# Confused about continuity of this function

• wumple
In summary, the conversation discusses the existence of solutions for the differential equation y'=1/(x+y) with a given initial condition, and the confusion surrounding the existence theorem and continuity of y' in the surrounding box. It is clarified that the existence theorem does not guarantee differentiability at a point where a solution exists.
wumple

## Homework Statement

For y'=1/(x+y), sketch a direction field and the solution through y(0)=0.

## Homework Equations

I'm confused as to why there is a solution through y(0) - I thought that the existence theorem says that if y' is continuous in a box, then there are solutions through all points in the box.

## The Attempt at a Solution

y' is not continuous in the box surrounding the y=-x line. So why does a solution exist there? Is y' actually continuous there? It approaches negative infinity from one side and positive infinity from the other.

"If P, then Q" is not the same thing as "If Q, then P".

So yes.. "if y' is continuous in a box, then there are solutions through all points in the box", but this does not mean "if a solution exists at point y(a)=b, then this solution is differentiable at a".

Take $$y=\sqrt{x}$$. y(0)=0 is defined, but y'(0) is undefined.

Another way of saying the same thing- "if P then Q" tells you what happens if P is false. It tells you nothing about what happens if P is false. In particular, it does not tell you that Q is false.

## 1. What is continuity of a function?

Continuity of a function refers to the property of a function where it has no sudden or abrupt changes in its output as the input changes. In other words, a function is continuous if it can be drawn without lifting the pencil from the paper.

## 2. How do you determine if a function is continuous?

A function is continuous if it satisfies three criteria: 1) the function is defined at the point in question, 2) the limit of the function exists at the point, and 3) the limit of the function equals the value of the function at that point.

## 3. Can a function be continuous at one point but not at another?

Yes, a function can be continuous at one point but not at another. A function is considered continuous at a specific point if it satisfies the three criteria mentioned above. If it fails to meet any of the criteria, then it is not continuous at that point.

## 4. What is the difference between continuity and differentiability?

Continuity and differentiability are two different properties of a function. Continuity refers to the smoothness of a function, while differentiability refers to the existence of the derivative of a function at a particular point. A function can be continuous but not differentiable, and vice versa.

## 5. How does continuity affect the behavior of a function?

Continuity is an important property of a function as it determines the behavior of a function at a specific point. If a function is continuous at a point, it means that it has no abrupt changes in its output at that point. This ensures that the function is well-behaved and predictable, making it easier to analyze and understand its behavior.

Replies
27
Views
2K
Replies
2
Views
743
Replies
10
Views
1K
Replies
6
Views
1K
Replies
8
Views
1K
Replies
39
Views
4K
Replies
6
Views
622
Replies
3
Views
1K
Replies
4
Views
3K
Replies
3
Views
1K