Recent content by ssayan3

Homework Statement Prove the function f(x,y) = x/y is continuous. As an added stipulation, the quotients of limits theorem may not be used. Homework Equations The Attempt at a Solution I have absolutely no idea how to go about this one. I can't even get a start on the scratchwork... Can...

Argh, that makes it even worse for me because now I really have no idea how to finish this one. What I have so far is: 1. Pick a point z in Compliment(S) 2.Then, z is not in S, and is not in Closure(S) by hypothesis

Oh! I'm sorry if I was unclear about something... "S\cupBdyS" refers to the union of the set S with its boundary, and is called Closure(S) It can also be referred to as the union of the interior of S with the boundary of S. (Someone tell me if I made an error!)

Homework Statement Hi guys, this problem gave me some trouble before, but I'd like to know if I have it worked out now... "If S = S\cupBdyS, then S is closed (S_{compliment} is open) Homework Equations S is equal to it's closure. The Attempt at a Solution 1. Pick a point p in...

Haha, well, during the five minutes after I wanted to ask you to give me a hint, I think i pretty much got it figured out... Forward direction: S is open relative to D if for any point p in S there exists delta >0 s.t. B(p,delta) intersect D is a subset of S 1. Since S is open relative to...

Conceptually, yes, I see what you are talking about. Can you give me another hint to push me along the path of proving it? Thanks for your help thus far

Homework Statement T is open relative to X iff for any p \in S there exists \delta > 0 such that B(p,\delta )\capX is \subset T Homework Equations T is open relative to X provided there exists an open subset U of R^n such that T = U\capX The Attempt at a Solution Okay, so, going...

Hm, I'm just not sure why I didn't think of that before! Thank you!

Homework Statement Using the epsilon-delta definition, prove that the function f:R^{2} \rightarrow R by f(x,y) = xy/((x^{2}) + (y^{2})), and f(0,0) = 0 is not continuous. The Attempt at a Solution I just really have no clue how to set up a delta-epsilon proof for functions that involve...

Haha, fantastic! Thanks to both of you. That makes things much easier to understand for me.

This is for an analysis class, so yes, I would think that I would have to use the definition of derivative in this one...

Homework Statement Prove that (ax^n)' = nax^n-1 using induction. I am very weak with induction proof, and I haven't had much trouble proving the basis step, but I can't seem to finish it... Homework Equations The Attempt at a Solution 1. Prove (ax)' = a (a(x+h) - a(x))/h =...

Hmm... if there were such a number z, then y could not be the least upper bound... Could the proof go something like this?: Choose arbitrary E>0, and let y be an upper bound of A Suppose z is an upper bound of A, and y>z>y-E. y is not the lub does this finish the proof?

Least Upper Bound proof... Homework Statement Suppose A is a nonempty set that has x as an upper bound. Prove that x is the least upper bound of the set A iff for any E>0 there exists a y in A such that y>x-E Homework Equations None The Attempt at a Solution The forward where you...

Hey guys, this isn't a math problem for homework or for a course, but simply for my own interest... This is one that I couldn't crack back in Analysis but that I'd like to get some help on resolving for my own peace of mind: Prove f(x,y) = xy is continuous at (x,y) in a domain D\subset R. I...