Your function split into two functions at x=0 only .. or at the point (0,0) .. ?!!
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