Articles of functions

Continuity of the Lebesgue function

If $x \in [0,1]$ has ternary expansion $(a_n)$, i.e. $x = 0.a_1a_2..$ with $a_n =0,1$ or $2$, define $N$ as the first index $n$ for which $a_n = 1$, and set $N = \infty$ if none of the $a_n$ are $1$ (when $x \in $ Cantor Set). Now, set $b_n = \frac{a_n}{2}$ for $n < […]

domain of composite function (is there a set rule)

In the function topic of “function combinations” or “function algebra”, for the basic arithmetic operations of the following: 1) f + g 2) f – g 3) f * g 4) f / g To find the domain of these, one needs to find the domains of each f, g and find the intersection to […]

How can I prove that the inverse function of a total recursive function is recursive?

The full question actually is : If $A\subseteq \mathbb{N}$ and $f: \mathbb{N} \rightarrow \mathbb{N}$ is a function. Prove that if $f$ is a total recursive function and $A\subseteq \mathbb{N}$ recursive, then $f^{-1}(A)$ is recursive. I guess that the part that $A\subseteq \mathbb{N}$ is recursive is helping us to understand that this statement is true. However, […]

Show the quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$ is strictly quasi-concave

I posted the question before show the quadratic function is quasi-concave. But I can not understand the only answer, and the author delated his account already. I rephrase my question and show my second attempt here. I have a quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$, with $x_i$ nonnegative and $A \in[0,1)$. And w.l.o.g. we […]

Having trouble showing continuity, and pointwise limit of this function.

The function that I am having trouble with is: $$ { f }_{ n }(x)=\begin{cases} 1\quad ,\quad x \in\{ 1,\frac { 1 }{ 2 } ,…,\frac { 1 }{ n } \} \\ 0\quad ,\quad otherwise \end{cases} $$ and in particular I am trying to show that the each $f_{n}$ is continuous at x = […]

Notation: is there a symbol for “not a function of”?

For example, let’s say a term $A(x,y)$, a function of two random variables $x$ and $y$, is the argument of an expectation over $y$. The resulting term is no longer a function of $y$. Is there a mathematical symbol that explicitly says this? I know there are workarounds–in this arbitrary case, bar notation and dropping […]

Is it always possible to transform one function into another?

Suppose f(x) = g(x) for all real x, and both f and g are sufficiently nice (perhaps we might limit them to be polynomials or analytic functions). Can we always manipulate one (with algebraic transformations) into the other? Is the same true for functions with multiple arguments? To say more precisely what I mean let […]

Commutative diagrams of sets and functions

Consider these two diagrams of sets and functions (with $f$ and $f’$ invertible): \begin{array}{ccccccccc} A & \overset{f}{\longrightarrow} & B && &B & \overset{f^{-1}}{\longrightarrow} & A\\ u\downarrow& & v\downarrow & &; & v\downarrow & & u\downarrow\\ A’ & \overset{f’}{\longrightarrow} & B’ & && B’ & \overset{f’^{-1}}{\longrightarrow} & A’ \end{array} Can I say that the first commutes […]

Finding all possible values of a Function

Let a function be defined as $f:N\to N$ and $x-f(x)= 19\left[\dfrac{x}{19}\right] – 90\left[\dfrac{f(x)}{90}\right] \forall x\in \Bbb N$ and $1900<f(1990)<2000$. Find all values of $f(1990)$. $$$$ I really have no clue as to how to go about this as I’ve never encountered such questions before. I would be truly grateful if somebody would please show me […]

A function $f(x)$ that increases from $0$ to $1$ when $x$ increases from $-\infty$ to $\infty$.

I am looking for a function $f(x) \in [0,1]$ when $x \in (-\infty, +\infty)$. $f(x)$ increases very fast when $x$ is small starting from $-\infty$, and then very slow and eventually approach $1$ when $x$ is infinity.