# Q in Continuity of functions of three var

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

mathman
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?

verty
Homework Helper
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.