1
vote
1answer
13 views

Creating switches for piece wise defined function?

How can I create "switches" [the term may be new, but I'll explain it] for piecewise defined functions ? Suppose a function: $$ f(x)=\begin{cases}\alpha\;,x\in D_1\\\beta\;,x\in ...
1
vote
3answers
52 views

Find a function that is surjective and not injective

Rules: No piecewise functions. The function must be even, odd, or both even and odd. It cannot be neither. If this is impossible, prove why. This is just something I came up with for fun while ...
2
votes
2answers
130 views

How can I get a good estimation of the following function

The function is $$ f(n) = \sum_{i=1}^{n} \frac{1}{2i-1}$$ How can I compute for example $f(20)$ or $f(50)$ without using a calculator. I want to have an approximation
1
vote
1answer
35 views

Characterization of nowhere differentiable functions

Let $N:=\{f\in C([0,1])\vert \text{ f is nowhere differentiable } \}$ and $A_n = \{f\in C([0,1]) \vert \exists x\in [0,1]s.t. \forall y\in[0,1]: |f(x)-f(y)|\leq n |x-y|\}$. Now I have already ...
2
votes
1answer
73 views

Functions which satisfy $\mathrm{f}(wz) =w\,\mathrm{f}(z)+z\,\mathrm{f}(w)$

Let $\mathrm{f}$ be a complex-valued function with the following property: $$\mathrm{f}(wz) =w\,\mathrm{f}(z)+z\,\mathrm{f}(w) $$ for all $w,z \in \mathbb C$. Necessary conditions are that ...
2
votes
2answers
70 views

A non-trivial, non-negative, function bounded below by its derivative with $f(0)=0$?

I did not know what to search to see if this existed elsewhere. But, I could not find it. Here's the question, does there exist a continuously differentiable function, $f: [0,1] \rightarrow ...
0
votes
2answers
91 views

function with restrictions in finding solutions

Please help... How to prove the following functional equation, which has no solution. $A + B + C + (A + B)C - 2 = 0$ has no solution, where $A = f(x)$, $B = g(x)$ and $C = h(x)$. Here $A, B$ and $C$ ...
6
votes
0answers
263 views

Can $ 262537412640768743.99999999999925 $ be beaten with simple expressions? [closed]

We know: $$\begin{align}e^{\pi \sqrt{163}} &= 262537412640768743.9999999999992500726\dots\\ x^{24} - 24&=262537412640768743.9999999999992511239\dots\end{align}$$ where $x$ is the real solution ...
3
votes
2answers
86 views

Number theory Exercise

for positive integer $n$, how can we show $$ \sum_{d | n} \mu(d) d(d) = (-1)^{\omega(n)} $$ where $d(n)$ is number of positive divisors of $n$ and $mu(n)$ is $(-1)^{\omega(n)} $ if $n$ is square ...
0
votes
0answers
114 views

a follow up question to the birthday-paradox question.

The previous question. My goal is to find a function of the difference (error) between F and G generally. F = $\Pi_{0}^{n-1}\frac{365-b}{365}$ G = $ \frac {364}{365}^{\frac {n^2-n}{2}}$ Now F ...
-1
votes
2answers
949 views

write text using an equation

Well like the batman equations and equations for heart, I once saw a site that draws equation for whatever text you type....but now I can't find it. Does anybody know such a site? Also a general ...
1
vote
1answer
82 views

Measuring how monotonically “staircase-like” a set of values is

A bit of a bizarre question here -- I'm looking for assistance in generating a robust metric to measure how monotonically "step-wise" a series of values is. The set must not start or end at a specific ...