I was trying to solve the practice problems in my textbook, but I am highly frustrated. The terrible thing is that my textbook has a few to no examples at all, just a bunch of theorems and definitions, so I have no idea how to solve real problems...I am feeling desperate... Note: Let x E R^n, f is continuous at a iff lim f(x) = f(a) x->a 1) Let f(x,y) = sin(xy) / x for x not=0. How should you define f(0,y) for y E R so as to make f a continuous function on all of R^2? Justify your reasonings. [I know that what I have to find is the limit of f(x,y) as (x,y)->(0,y) and set this limit value equal to f(0,y), but my trouble is that I have no idea how to evaluate limit of f(x,y) as (x,y)->(0,y)] 2) Let f(x,y) = (xy) / (x^2 + y^2). Show that, although f is discontinuous at (0,0), f(x,a) and f(a,y) are continuous functions of x and y, respectively, for any a E R (including a = 0). We say that f is "separately continuous" in x and y. [f(x,a) = f(x) = (ax) / (x^2 + a^2) If "a" not =0, f(x) is a rational function, so it's continuous everywhere in its domain, i.e. for all x E R If a=0, f(x) = 0/x^2, so f(x) is discontinuous at x=0 This is what I've done so far, and this is driving me crazy, especially in the last line where I got that for a=0, f(x) is discontinuous at x=0. So I must have done something wrong, but I have no idea where...] 3) Let f(x,y) = y(y - x^2) / x^4 if 0 < y < x^2, and f(x,y)=0 otherwise. Find ALL point(s) for which f is discontinuous. [This looks really hard. How can I approach this problem? Any help?] Any help is greatly appreciated!