Is the Function F(x, y) Continuous at All Points?

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In summary: If you can help me out that would be great! thank you!In summary, F is continuous in each variable separately, but is not continuous overall.
  • #1
tomboi03
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Let F: R x R -> R be defined by the equation
F(x x y) = { xy/(x^2 + y^2) if x x y [tex]\neq[/tex] 0 x 0 ; 0 if x x y = 0 x 0
a. Show that F is continuous in each variable separately.
b. Compute the function g: R-> R defined by g(x) = F(x x x)
c. Show that F is not continuous.

I know how to do part a...
but I'm not sure how to do b or c.

If you can help me out that would be great! thank you!
 
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  • #2
Well, b seems rather straightforward, just plug it in.

For c, you could show that there is a point for which the limit value depends on the path you take. For example, showing that
[tex]\lim_{x \to 0} F(x, 0) \neq \lim_{y \to 0} F(0, y)[/tex]
would prove that F is not continuous at (0, 0) because then it shouldn't matter how you get to (0, 0). I think that b should give you a hint on which point and paths to consider :)
 
  • #3
Sorry to rehash something so old; I was doing a search for the general situation;
wonder if someone knows the answer:

An important/interesting question would be if we can add some new condition
so that if f(x,.) and f(.,y) are continuous, then so is f(x,y).

For one thing, the continuity of maps f:XxY-->Z is often used in constructing
homotopies; I have never seen the issue of why/when these homotopies are
continuous.
 
  • #4
Sorry, I can't access the 'Edit' button for some reason.

A standard counter to having a function beeing continuous in each variable, yet
not overall continuous is the one given by tomboi03.

My point is that a homotopy between functions f,g, is defined to be a _continuous_ map H(x,t) with H(x,0)=f and H(x,1)=g. Since we cannot count on H(x,t) being continuous when each of H(x,.) and H(.,y) is continuous :what kind of result do we use to show that our map H(x,t) is continuous? Do we use the 'good-old' inverse image of an open set is open , or do we use the sequential continuity result that [{x_n}->x ] ->[f(x_n)=f(x)]
(with nets if necessary, i.e., if XxI is not 1st-countable)?

I saw a while back an interesting argument that if continuity on each variable alone
was enough to guarantee continuity, then every space would have trivial fundamental group:

e.g, for S^1, use H(e^i*2Pi*t,s) :=e^i2Pi(t)^s
 
  • #5


a. To show that F is continuous in each variable separately, we need to show that the limit of F as x or y approaches any value is equal to the value of F at that point. This can be done by using the definition of continuity and evaluating the limit at different points. For example, for x, we can fix y at a constant value and take the limit as x approaches that value. Similarly, for y, we can fix x at a constant value and take the limit as y approaches that value. If the limit is equal to the value of F at that point, then F is continuous in that variable.

b. To compute g(x), we substitute x for both x and y in the equation for F. This gives us g(x) = x^2/(x^2 + x^2) = x^2/2x^2 = 1/2. So g(x) is a constant function, equal to 1/2 for all values of x.

c. To show that F is not continuous, we need to find a point where the limit of F is not equal to the value of F at that point. We can choose the point (0,0) since it is the only point where F is not defined. Taking the limit as x and y approach 0, we get F(0,0) = 0, but the limit is equal to 1/2, as shown in part b. Therefore, F is not continuous at (0,0) and is not continuous overall.
 

Related to Is the Function F(x, y) Continuous at All Points?

1. What is continuity in scientific terms?

In scientific terms, continuity refers to the smooth and uninterrupted flow of a physical or mathematical quantity or process. In other words, it means that there are no abrupt changes or discontinuities in the behavior of a system or function.

2. How can you prove that F is not continuous?

To prove that F is not continuous, we can show that there is at least one point in its domain where the limit of the function does not equal the function value at that point. This would violate the definition of continuity, which requires that the limit and function value be equal at all points in the domain.

3. What is the difference between point discontinuity and jump discontinuity?

Point discontinuity occurs when the limit of a function does not exist at a specific point in its domain. This can happen when there is a hole or gap in the graph of the function. Jump discontinuity, on the other hand, occurs when the limit from the left and the limit from the right exist, but they are not equal. This creates a sudden jump or break in the graph of the function.

4. Can a function be continuous at some points and discontinuous at others?

Yes, a function can be continuous at some points and discontinuous at others. This is known as a piecewise continuous function, where the function has different rules or behaviors in different sections of its domain. In order for a function to be considered continuous, it must be continuous at all points in its domain.

5. How does continuity relate to differentiability?

Continuity and differentiability are closely related concepts. A function is differentiable at a point if it is continuous at that point and has a well-defined derivative. This means that a function cannot be differentiable at a point if it is not continuous at that point. However, a function can be continuous at a point without being differentiable at that point, as in the case of a sharp turn or corner in the graph of the function.

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