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 May19 revised How to solve this integral easily: $\int \frac{x\cdot \sqrt[3]{x+2}}{x+\sqrt[3]{x+2}} dx$added dx to the integrand May19 comment Why is $\varphi$ called “the most irrational number”?http://www.ams.org/samplings/feature-column/fcarc-irrational1 May19 suggested suggested edit on How to solve this integral easily: $\int \frac{x\cdot \sqrt[3]{x+2}}{x+\sqrt[3]{x+2}} dx$ May9 awarded Notable Question Mar23 comment $\sqrt{\frac{a}{2}}+\sqrt{\frac{a}{2}}\ge \sqrt{a}$?Or you could just end with $\sqrt{2}\sqrt{a}=\sqrt{2a} > \sqrt{a}$ instead of having to explain that $\sqrt{2}>1$. Mar23 comment $\sqrt{\frac{a}{2}}+\sqrt{\frac{a}{2}}\ge \sqrt{a}$?This is good informal reasoning but not a good proof. If you start out by assuming a conjecture is true and then arrive at a tautology, that does not necessarily prove that the conjecture is true. It just shows that the consequence of it being true is a tautology. Mar23 suggested suggested edit on $\sqrt{\frac{a}{2}}+\sqrt{\frac{a}{2}}\ge \sqrt{a}$? Mar6 comment How to understand why $x^0 = 1$, where $x$ is any real number?The idea that exponentiation can be represented as iterated (repeated) multiplication really only holds for natural number exponents. Think about negative exponents: what is 5 times itself, negative 2 times? It doesn't make sense in our minds, even though we know the value of $5^{-2}$. Even worse is with rational, and worse, irrational, exponents. Mathematically we have a way of calculating them, but we don't tend to think of them as iterated multiplication. This concept breaks down when we get to 0-valued exponents as well. Mar4 revised name this functionadded 146 characters in body Mar4 revised name this functionadded 146 characters in body Mar4 comment name this functionI like your ideas but they're all piecewise. Is there one not piecewise? Is there one that has infinitely many derivatives (continuous everywhere or perhaps even uniformly continuous)? Mar4 revised name this functionadded 146 characters in body Feb11 revised Fibonacci numbers that are powers?edited body Feb11 comment name this functionI've tried these, they do not fit all the criteria. Neither the logistic function nor the erf function meet both $f(1)=.5$ and $f'(0)=1$. (the erf function is the integral of the Gaussian curve you mentioned in your post) Feb11 comment name this functionbecause $\frac{d}{dx} \left(\frac{2}{\pi} \arctan(x)\right) \ne 1$ at $x=0$ Feb11 comment name this functionInteresting: $f'$ is bounded by $1$, $f''$ is bounded by $2$, $f'''$ is bounded by $6$, $f^{\mathrm{iv}}$ is bounded by $24$. Is $f^{\mathrm{n}}$ always bounded by $n!$? Feb11 asked name this function Feb10 asked Fibonacci numbers that are powers? Feb10 revised which is the cardinal of primes?simplified hyperlink, changed Aleph to Aleph-naught Feb10 comment which is the cardinal of primes?Just because there is no formula for a bijection does not mean there isn't a bijection. The common misunderstanding that a lot of people have is that functions need to have "formulas" or "equations". This is not always the case. Check out the Dirichlet Function.