TITLE: Explanation regarding continuity in piecewise function
QUESTION [0 upvotes]: Let $ \begin{equation}
     f(x) = \begin{cases}
          e^{-1/x}, & x > 0,\\
          0, &  x \le 0
          \end{cases} \end{equation}$
I have proven that $f(x)$ is continuous at $x \in R\setminus {0}$, and in order to prove that it is continuous at $x=0$, consider  $(x_{n}) \in \mathbb R_{+}$ as well as $(y_{n}) \in \mathbb R_{-}$
For a sequence $x_{n} \to 0, n \to \infty$, we have $f(x_{n})=e^{-1/x_{n}}$. My intuition would be to just say
$\lim_{n \rightarrow \infty} f(x_{n}) = \lim_{n \rightarrow \infty} \frac {1}{e^\frac{-1}{x_{n}}}$=$\frac{1} {e^\lim {n \rightarrow \infty}{\frac{-1}{x_{n}}}}=0$.  But would this not be 'illegal' as taking the limit into the equation already constitutes it being continuous at $0$.   Isn't that exactly what I am attempting to prove?
I'm struggling to understand this nuance. Any help would be greatly appreciated

REPLY [1 votes]: What you've done is not illegal. You did not take the limit of your function, you've taken the limit of the function $f(x)=e^{-\frac{1}{x}}$, which happens to coincide with your function on $(0, \infty)$. Your function and $e^{-\frac{1}{x}}$ have the same values, therefore they have the same limit when $x\to 0^{-}$.
(Of course, you now need to prove that $\lim_{x\to 0^{-}}e^{-\frac{1}{x}}=0$, which is more-or-less done in your question; I am not sure if your write-up is rigorous enough, but this is another matter...)


TITLE: Teaching prime number theorem in a complex analysis class for physicists
QUESTION [17 upvotes]: This is a question about pedagogy.
I want to sketch the proof of the prime number theorem or any other application of complex analysis to number theory in a single lecture, in a complex analysis course for physics students.
Is this possible? How would you go about it?

REPLY [1 votes]: You can start by defining the Riemann Zeta function as $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$ for $s$ real and then prove the Euler product formula, $\zeta(s) = \prod_{p} \frac{1}{1-p^{-s}}$ for $s>1$ where the product runs over all prime numbers.  Then you can view the zeta function as a function of a complex variable.  This will illustrate a nontrivial connection between a complex function and the prime numbers.  After this, I suggest that you go through Riemann's epoch 8 page paper with the class as outlined in Edwards book on the zeta function.  Here, Riemann uses tools from complex analysis such as contour integration to study zeta.  He proves analytic continuation, the functional equation for zeta, and goes on to examine the connection between the zeros of the zeta function and the distribution of the prime numbers.