In whatever little I have learnt about calculus of two variable functions I have been having some serious problems in the way continuity of a function is defined.(adsbygoogle = window.adsbygoogle || []).push({});

We say that a function is not continuous if we can find two paths of approach along which the value of the independent variable is different.Now if I have to find such paths leading to a say a point,say the origin(0,0,0) the simplest way I can think of is to put the function equal to some constant c and obtain a contour curve for that value.Now in the neighborhood of the point under consideration(origin here),if I can limit the z-value of the contour lines crossing the neighborhood :

f(0,0)-ε< c_{ε }<f(0,0)+ε

where ε>0 is an arbitrary number,I can say that the function is continuous at that point.This follows from the formal definition of continuity and is easily understood.

But I will put forward two examples:

1.f(x,y)=x^{3}+y^{3}/x-y

This function is not continuous at (0,0).A few contour lines for z={1,5,3} are here:

http://rechneronline.de/function-graphs/" [Broken]

By this I can conclude that the function is non-continuous at (0,0).

2.f(x,y)=xy/√ (x^{2}+y^{2})

This function comes out to be continuous at (0,0) by the formal δ-ε method.However the corresponding contour lines look like this:

http://rechneronline.de/function-graphs/" [Broken]

Now I have paths to (0,0) that have different z-values even though the function is continuous at

(0,0).Can somebody help me?

Edit:The graphs linked are not working.Please use these functions:

Function 1:

1.x^3+y^3-x+y

2.x^3+y^3-3*x+3*y

3.x^3+y^3-5*+5*y

Function 2:

4.(x*y)-sqr(x^2+y^2)

5.(x*y)-3*sqr(x^2+y^2)

6.(x*y)-3*sqr(x^2+y^2)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Continuity for Two Variable Function

**Physics Forums | Science Articles, Homework Help, Discussion**