Continuity of a function of 2 variables

Mar 2010
16
1
Hi, can someone just tell me if my argument for this question is correct (and if it isn't, what is the correct answer)

Is the function f : R^2 -> R defined as f(x,y) = {(sin xy)/x for nonzero x
y for x=0 }

continuous at (0,0) ?

My answer: If we take any two sequences of real numbers a_n and b_n that tend to zero as n to infinity, then we have that (a_n, b_n) -> (0,0) and f(a_n, b_n) = sin(a_n b_n)/a_n and since for small values of x, sin(x) is very close to x we have that f(a_n, b_n) has the same limit as a_n b_n / a_n = b_n which goes to 0 = f(0,0). Thus it is continuous at (0,0).

Thanks :)
 
Jan 2010
564
242
Kuwait
Your function split into two functions at x=0 only .. or at the point (0,0) .. ?!!
 
Mar 2010
16
1
Your function split into two functions at x=0 only .. or at the point (0,0) .. ?!!
What do you mean? It is (sin xy)/x when x is not 0, and y when x is 0. So for example f(0,3) = 3 ... f(0,0) = 0 ..... f(2,3) = (sin 6)/2