Continuity of Functions Problem

johnners

Hi I came across this problem I found rather difficult and I was hoping someone could help me in the right direction

Give an example of a function f : R^2 ->(arrow) R that has the following
properties:
f is not continuous at (0, 0).
If g(t) = f(t, t) then g is continuous at 0.
If h(t) = f(t, t2) then g is continuous at 0.

Thanks

ZorbaTehZ

By t2 do you mean [latex]t^2[/latex] or something else? Anyway, the idea is with these types of questions is to take some function f(x,y) and then to look at planes through the z-axis - an easier question to get an idea is: construct f(x,y) so that f(t,t) is discontinuous at 0 but continuous along the line of the x-axis or y-axis, then f(x,y)=xy/(x^2+y^2) if (x,y)!=0 and f(x,y)=0 if (x,y)=0 works, and it should be fairly clear why.

(Actually just realised that the f(x,y) I quoted is continuous along any line parallel to the x-axis or the y-axis, so maybe there's a simpler example)

Michael Collins

johnners wrote: »

Hi I came across this problem I found rather difficult and I was hoping someone could help me in the right direction

Give an example of a function f : R^2 ->(arrow) R that has the following
properties:
f is not continuous at (0, 0).
If g(t) = f(t, t) then g is continuous at 0.
If h(t) = f(t, t2) then g is continuous at 0.

Thanks

What you need is a function of two variables, f(x,y) that is continuous, at the point (0,0), along both the line x=y and the curve x=y^2 in the xy-plane, but not continuous along some other curve. (I'm assuming you meant "then h is continuous at 0" in the last line above.)

Here's an example of a function that is continuous at (0,0) along the line x=y, but along no other line at the origin:

[latex] f(x,y) = \left\{\matrix{1\hbox{ for } x=y\cr0\hbox{ otherwise}}\right.[/latex]

Try to generalise this for your case.

Blackpanther95

Haha some1 being cheating on their analysis assignment tut tut