Continuity and Polar Coordinates

[schmidt7100]

Homework Statement

Show that the function f(x,y)= xy/sqrt(x^2+y^2) is continuous at the origin using polar coordinates. f(x,y)=0 if (x,y)=(0,0)

Homework Equations

r=sqrt(x^2+y^2)
x=rcos(theta)
y=rsin(theta)

The Attempt at a Solution

So, converting this equation to polar coordinates, I get rsin(theta)cos(theta). However, after this I'm stumped as to how I prove that this is continuous at the origin.

 

[rock.freak667]

At the origin in polar coordinates (r,θ), shouldn't r=0 and θ=0 ?

 

[HallsofIvy]

Not exactly- at the origin r= 0 and $\theta$ is undefined.

However, Schmidt7100, the point is that r alone measures the distance of a point from the origin- $\theta$ is irrelevant. That means that, if $\lim_{r\to 0}f(r,\theta)$ does not depend on $\theta$, then that is the limit of $f(r, \theta)$ as $(r, \theta)$ goes to the origin.

Note that is NOT the case for Cartesian coordinates. Since the distance from (x, y) to (0, 0) depends upon both x and y. I am sure you have seen examples of functions in (x, y) that give different limits as you go to (0, 0) along different paths.

 

[hunt_mat]

You don't have to use polar co-ordinates, let
$$x_{n}=\frac{1}{n},\quad y_{n}=\frac{1}{n}$$
Insert these into your equation for f and let $$n\rightarrow\infty$$, compute the limit and if it's the same as f(0,0) you've proved continuity.