Tagged Questions

29
votes
4answers
744 views

When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?

The question is: When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution? Edit I should ...
1
vote
1answer
78 views

Use the Mean Value Theorem to prove the Binomial Inequality?

Binomial Inequality: $\forall x \in \mathbb{R}, x\geq-1, \forall n \in \mathbb{N}: (1+x)^{n} \geq 1+nx$ I need to prove this using the Mean Value Theorem: If $f$ is continuous on $[a,b]$ and ...
1
vote
2answers
51 views

product rule for matrix functions?

Given a real rectangular matrix $X$, and two scalar-valued matrix functions, $f(X)$ and $g(X)$, does the product rule for differentiation of a product of scalar valued functions, hold when ...
1
vote
3answers
101 views

confusion regarding the concept of a function

I was reading principles of mathematical analysis by Walter Rudin chapter 2 when a confusion about the definition of a function cropped up. (Read definition in comment below) I had thought that ...
2
votes
2answers
129 views

Function whose integral is equal to the function

Let $\lambda \in \mathbb{R}$ be a constant, $\lambda \neq 0$. Is there a function $f \in C[0,1]$, $f \neq 0$, that satisfies the following relation: $$\lambda f(s) = \int_0^s f(t) \, dt$$ Attempt at ...
1
vote
1answer
96 views

Is the inverse function continuous?

Suppose $f:X\rightarrow Y$ is 1-1 and continuous. Is $f^{-1}:f(X)\rightarrow X$ continuous too? If not, can you explain it?
7
votes
6answers
1k views

Example where $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective

Can anyone come up with an explicit example of two functions $f$ and $g$ such that: $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective? I tried the following: $$f:\mathbb{R}\rightarrow ...
15
votes
3answers
528 views

What kind of “mathematical object” are limits?

When learning mathematics I tend to try to reduce all the concepts I come across to some matter of interaction between sets and functions (or if necessary the more general Relation) on them. Possibly ...
2
votes
1answer
138 views

Derivative of an indicator inside an integral

I have a fairly basic question that relates to understanding a particular derivation. I have the following function $Q(x) = E\left[I(F(x+\varepsilon)>c)\right]$, where $x \in R$, $\varepsilon ...
1
vote
1answer
110 views

Inequality holds?

Can anyone prove that $$ \frac{\sum\limits_{i=1}^{k*} a_i i (x+\epsilon)^{(i-1)}}{\sum\limits_{i=1}^{k*} a_i (x+\epsilon)^{i}+k^*-1}>\frac{\sum\limits_{i=1}^{k*} a_i i ...
2
votes
1answer
114 views

Antisymmetric functions in higher dimensions

For an antisymmetric function $f:\mathbb R\rightarrow \mathbb R$ (i.e. f(x)=-f(-x)) we have: necessary condition for the differential of f of order $r$ to not vanish at $0$ is that $r$ is odd. My ...
0
votes
1answer
64 views

finding the function

Define a function $f\colon\mathbb{R}\to\mathbb{R}$ which is continuous and satisfies $$f(xy)=f(x)f(y)-f(x+y)+1$$ for all $x,y\in\mathbb Q$. With a supp condition $f(1)=2$. (I didn't notice that.) How ...
3
votes
2answers
108 views

Sequence of functions (convergence)

Let be $f(x)=\frac{2x}{1+x} $ function and $ x_0 > 0 $. With the help of this, form the $x_{n+1}=f(x_n)$ sequence. Is $x_n$ convergent and if yes what is the limit? Thank you very much in advance! ...