Intereting Posts

Improper integral of $\sin^2(x)/x^2$ evaluated via residues
Derive the centroid of an area from a limiting procedure
The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$
What are some counter-intuitive results in mathematics that involve only finite objects?
What is needed to make Euclidean spaces isomorphic as groups?
Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
Picking random points in the volume of sphere with uniform probability
Is $e^{n\pi}$ transcendental?
Cancellation Law for External direct product
How to define multiplication in addition terms in monadic second order logic?
Let $f$ be differentiable on $(0,\infty)$.Show that $\lim\limits_{x\to \infty}(f(x)+f'(x))=0$,then $\lim\limits_{x\to \infty}f(x)=0$
Annihilators in matrix rings
Prove:$A B$ and $B A$ has the same characteristic polynomial.
Applications of Weierstrass Theorem & Stone Weierstrass Theorem
Proving that product of two Cauchy sequences is Cauchy

We usually only see the graph $y=x^x$ for $x>0$, because $x^x$ is a complex number for most negative values of $x$. Yet here is a full graph of $y=x^x$ on the real line:

This graph may seem like it’s not even a function, failing the vertical line test, but what’s actually going on is that the graph contains infinitely many holes. It’s true that $x^x$ is not a real number for almost all values of $x<0$, but it is real in the rare situation when $x$ is a rational number which has an odd denominator when written in simplest form. In that case, $x^x$ is a positive real number when $x$ can be written as an even number divided by an odd number, and a negative real number when $x$ can be written as an odd number divided by an odd number. So just those rational points are being graphed, but since the rational numbers with odd denominators are dense in the real numbers it looks like we have continuous curves.

My question is, what are the continuous curves that seem to be there? Any continuous curve defined on a dense subset of the reals can be uniquely extended to a continuous function on all the reals. So what continuous function passes through all the points $(x,x^x)$ where $x<0$ and $x$ can be written as an even number divided by an odd number? (I’d ask the analogous question about the second curve, but it seems to be a mirror image of the first.)

- Prove partial derivatives of uniformly convergent harmonic functions converge to the partial derivative of the limit of the sequence.
- showing $-\eta(s) = \lim_{z \ \to \ -1} \sum_{n=1}^\infty z^{n} n^{-s}$
- show that function $f(z)=(z^2+3z)/(e^z-1)$ can be expressed as a power series
- Evaluate a definite integral involving Airy functions
- Residue at infinity (complex analysis)
- Proving $\left(\sum_{n=-\infty}^\infty q^{n^2} \right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi \tau)}$

Any help would be greatly appreciated.

Thank You in Advance.

- Can two analytic functions that agree on the boundary of a domain, each from a different direction, can be extending into one function?
- Finding the Laurent series of $f(z)=\frac{1}{(z-1)^2}+\frac{1}{z-2}$?
- A question regarding power series expansion of an entire function
- Proving $|f(z)|$ is constant on the boundary of a domain implies $f$ is a constant function
- An inequality on holomorphic functions
- Principal part of Laurent expansion.
- On continuity of roots of a polynomial depending on a real parameter
- Holomorphic function has at most countably zeros
- Simplify $Im \left(\frac{az+b}{cz+d}\right)$
- Proving theorem connecting the inverse of a holomorphic function to a contour integral of the function.

The continuous curves that pass through the points satisfying $y = x^x$ on $x \in (-\infty, 0)$ are simply $$g(x) = \pm \frac{1}{|x|^{-x}}.$$

- A “reverse” diagonal argument?
- On $x^3+y^3=z^3$, the Dixonian elliptic functions, and the Borwein cubic theta functions
- What type of discontinuity is $\sin(1/x)$?
- References about Iterating integration, $\int_{a_0}^{\int_{a_1}^\vdots I_1dx}I_0\,dx$
- Residue at infinity (complex analysis)
- Hardy's inequality again
- For any integer n greater than 1, $4^n+n^4$ is never a prime number.
- Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$
- Geometric interpretation of mixed partial derivatives?
- Dirichlet Function Pointwise Convergence
- Is the real number structure unique?
- Is 0.9999… equal to -1?
- How many ordinals can we cram into $\mathbb{R}_+$, respecting order?
- Show that $L^1$ is strictly contained in $(L^\infty)^*$
- Prove that if $\phi'(x) = \phi(x)$ and $\phi(0)=0$, then $\phi(x)\equiv 0$. Use this to prove the identity $e^{a+b} = e^a e^b$.