# Q in Continuity of functions of three var

• shnayzr
However, it is clear that "inside" means the points inside the paraboloid, and "outside" means the points outside the paraboloid.
shnayzr
Hello everyone.

Last week I had an exam in advanced calculus. One of the questions asked about the continuity of a function of three variables.

However, the doctor gave me 0 out of 3 for the question while i am sure that my answer is correct and i told him that but he insisted that its wrong.

can you help me?Question:

Discuss the continuity of the function f(x,y,z)=8xy/(x^2+y^2-z)

-------------------------------

the function is continuous in all the points except the points lying on the
circular paraboloid z=x^2 + y^2

-------------------------------

the domain of f(x,y,z) is all the point lying inside or outside (but not on) the
circular paraboloid z=x^2 + y^2

the function is continuous in all the points that lie inside or outside the
circular paraboloid z=x^2 + y^2
-------------------------------Thank you very much

shnayzr said:
Hello everyone.

Last week I had an exam in advanced calculus. One of the questions asked about the continuity of a function of three variables.

However, the doctor gave me 0 out of 3 for the question while i am sure that my answer is correct and i told him that but he insisted that its wrong.

can you help me?

Question:

Discuss the continuity of the function f(x,y,z)=8xy/(x^2+y^2-z)

-------------------------------

the function is continuous in all the points except the points lying on the
circular paraboloid z=x^2 + y^2

-------------------------------

the domain of f(x,y,z) is all the point lying inside or outside (but not on) the
circular paraboloid z=x^2 + y^2

the function is continuous in all the points that lie inside or outside the
circular paraboloid z=x^2 + y^2
-------------------------------

Thank you very much

It looks like he was somewhat picky.

The first sentence says there are points in (x,y,z) space not in the domain. You need a very precise definition of domain to exclude these points and he may feel that they shouldn't be excluded.

The second sentence needs a definition for inside and outside. This surface doesn't enclose anything, so there is no inside or outside.

Just to play devil's advocate ...

1. "Inside the paraboloid" could be interpreted as meaning "on the paraboloid", especially, I would imagine, by someone for whom English is a second language.

2. You technically didn't say where the function was discontinuous.

I agree, though, that the grading seems a bit picky. Did he give an explanation for why he is saying your answer is wrong?

mathman said:
It looks like he was somewhat picky.

The first sentence says there are points in (x,y,z) space not in the domain. You need a very precise definition of domain to exclude these points and he may feel that they shouldn't be excluded.

The second sentence needs a definition for inside and outside. This surface doesn't enclose anything, so there is no inside or outside.

I doubt the domain is the issue; by not specifying the domain, one should choose the largest domain possible. And, as a rule (I hope), a function should always be total, this is very widely assumed and should be the standard. Recall that the domain is also called the domain of definition. So, the function is not continuous on the paraboloid because it is not defined on the paraboloid, but it is continuous at all points in the domain.

I agree with your second point, "inside" and "outside" may not have been defined.

## 1. What is "Q" in Continuity of functions of three var?

"Q" refers to the set of rational numbers, which are numbers that can be expressed as a ratio of two integers. In the context of Continuity of functions of three variables, it represents the values that the function can take on along the x, y, and z axes.

## 2. How is Continuity of functions of three var defined?

Continuity of functions of three variables is the property of a function where the limit of the function at any point in its domain is equal to the value of the function at that point. In other words, there are no sudden jumps or breaks in the function's graph.

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

Continuity refers to the smoothness of a function, where there are no sudden jumps or breaks in the graph. Differentiability, on the other hand, refers to the existence of a derivative at a certain point on the function's graph. A function can be continuous but not differentiable, and vice versa.

## 4. How is the Continuity of a function of three var tested?

The Continuity of a function of three variables can be tested using the three-dimensional version of the epsilon-delta definition. This involves taking the limit of the function at a given point and ensuring that it is equal to the value of the function at that point. If this condition is met for all points in the domain, the function is continuous.

## 5. What are some real-world applications of Continuity of functions of three var?

Continuity of functions of three variables has many real-world applications, such as in physics, engineering, and economics. For example, it can be used to model the behavior of fluids, the motion of objects in three-dimensional space, and the optimization of multi-variable systems. It is also used in computer graphics to create smooth and realistic images and animations.

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