Recent content by Usagi

Show, using the $\epsilon-\delta$ definition of continuity, that the modified Dirichlet function, i.e., $f(x) = x$ if $x$ is rational and $f(x) = 0$ if $x$ is irrational, is discontinuous at all points $c \neq 0$ My attempt: Is the following argument right (using the sequential definition of...

Ah yup, makes perfect sense, thank you!

I understand this intuitively, but how can I make it rigorous using $\epsilon-\delta$ arguments?

Thanks! However, how do you that this neighbourhood is contained in (a,b)?

Let $a$, $b$ be reals and $f: (a,b) \rightarrow \mathbb{R}$ be twice continuously differentiable. Assume that there exists $c \in (a,b)$ such that $f(c) = 0$ and that for any $x \in (a,b)$, $f'(x) \neq 0$. Define $g: (a,b) \rightarrow \mathbb{R}$ by $\displaystyle g(x) = x -\frac{f(x)}{f'(x)}$...

Yup that is one way, however I was wondering whether my proof using the sequential characterisation is correct?

Show that $\lim_{x \rightarrow 0} \frac{1}{x}$ does not exist. Is my following argument correct? I will show there exists a sequence $(x_n) \subset \mathbb{R} \backslash \{0\}$ satisfying $x_n \neq 0$ and $(x_n) \rightarrow 0$, but $\lim_{n \rightarrow \infty} f(x_n)$ does not exist. Consider...

I can confirm I didn't miss anything. I've copied the q exactly as it is. Also i think part a) is harder than it seems. This is my working so far: So, define $U_k$ as the random variable that denotes the amount of petrol that car $k$ fills, $k = 1, 2, 3, \cdots$. Thus, $U_k, k = 1, 2, 3...

Thanks, how about part (a)?

Question: A single-pump petrol station is running low on petrol. The total volume of petrol remaining for sale is 100 litres. Suppose cars arrive to the station according to a Poisson process with rate \lambda, and that each car fills independently of all other cars and of the arrival...

I am self studying real analysis and I am doing an exercise which is proving the Cantor-Bernstein Theorem. **Question:** Assume there exists a 1-1 function $f:X \rightarrow Y$ and another 1-1 function $g:Y \rightarrow X$. Follow the steps to show that there exists a 1-1, onto function $h:X...

Awesome, thanks Opalg, I had a feeling the initial question had a mistake in it :)

Thanks, Yup I did that however how does it simplify down the RHS to equal the LHS?

http://img546.imageshack.us/img546/3171/integralbo.jpg For the above expression, I was told that it can be proven using Green's Theorem on the line integral on the RHS, however I can't seem the prove the equality. Note that $G$, $H$, $f$ are functions of $x_1$ and $x_2$. So I apply Green's...

Thanks Ackbach, I've had a read of Troutman's book, it is indeed very insightful however there isn't much on integral constraints and optimisation of multi-integral functions, do you have any ideas how to solve the above optimization problem? Cheers.