Recent content by jdz86

Homework Statement Use residues to evaluate the improper integral: \int^{\infty}_{- \infty} \frac{cos(x)dx}{(x^{2} + a^{2})(x^{2} + b^{2})} = \frac{\pi}{a^{2} - b^{2}} ( \frac{e^{-b}}{b} - \frac{e^{-a}}{a} ) Homework Equations a>b>0 The Attempt at a Solution If someone could...

Well if the function is non-zero at every point and continuous, then when taking the value of the function at some point along a given interval, that point will be non-zero as well since every other point along the function is non-zero. Is that what you were asking??

Haha, I must've just stared at your statement for about 2 minutes, then it clicked. So f would have to be zero everywhere because if it was greater than zero than it's integral would have to be greater than zero as well. Thanks. Now I just have to put everything together.

With your example f(x)>d/2, since f(c)=d where you have x=c so then f(x)=d. And d/2<d. But I still can't see the relation to my original problem.

And the answer would be no, because all points in the circle are strictly 2. Thanks for the help. Didn't even think to look at open balls in a plane. I'll use the same notions to show (b) as well, thanks.

O wait, for (a) is it not continuous because taking some ball around the boundary you get two different values for the function in that one ball, and given some epsilon the values in the ball might differ from x by a value greater than epsilon. Does that have any relevance??

f converges to c??

Well it is the boundary of the unit circle. On the boundary f(x) = 0, is that why it isn't continuous??

Homework Statement Let f(x,y) = { 2 if x^{2}+y^{2} < 1 , and 0 otherwise Using the definition of continuity to show that: (a) f is not continuous at each point (x_{0},y_{0}) such that x^{2}_{0} = y^{2}_{0} = 1 (b) f is continuous at all other points (x_{0},y_{0}) in the plane...

Well the upper bound, sup(f) = b , and the lower bound, inf(f) = a And with Riemann Sums, I don't see how you can add up the partitions and take the Upper Sum and Lower Sum of a function that you don't know. I just can't see where they fit into the proof.

Homework Statement Let f : [a,b] \rightarrow \Re be continuous and assume f \geq 0. Prove that if \int_{[a,b]}f = 0 then f = 0. Homework Equations Nothing really. If relevant, mean value theorem was discussed in earlier problems, so I'm not sure if it fits though. The Attempt at a...

yep, definitely wrote it wrong, (a) was what was given, thought it was a question. the question was something like this: using what was given, graph each of the following on the given axis, f(x),g(x), f \wedge g, f \vee g: f(x)=sinx, g(x)=cosx, x in [0,2pi] and graph f(x)=x(x-1)(x-2)(x-3)...

Homework Statement (a) Let f,g: [a,b] \rightarrow \Re. Define: f \vee g(x) = max(f(x),g(x)), x\in [a,b] f \wedge g(x) = min(f(x),g(x)), x\in [a,b] (b) Let f_{+} = f\vee0, f_{-} = -(f\wedge0) Show that: f = f_{+} - f_{-} abs value of f = f_{+} + f_{-}...

got it, thanks guys. where n=3 the lower partition would be 0+1/9+2/9 = 1/3 and the upper partitions would be 1/9+2/9+1/3 = 2/3. then for the n case, in my question, the sum of the upper partitions would be 1/n^2 + . . . + n/n^2 where the sum of 1 to n = n(n+1)/2, and plugging in you get n+1/2n...

Homework Statement Let f(x) = x, x \in [0,1], P_{n} = {0, \frac{1}{n}, \frac{2}{n},..., \frac{n}{n} = 1}. Calculate U_{P_{n}}(f) and L_{P_{n}}(f). Homework Equations U_{P_{n}}(f) is the sum of the upper partitions and L_{P_{n}}(f) is the sum of the lower partitions. A hint was...