0
votes
1answer
18 views

Are all singular functions of bounded variation?

Let $f$ be a function of bounded variation on $[a,b]$. Then there exist a unique pair (up to adding a constant) of absolute continuous function $g$ and singular function $h$ (i.e., $h'=0$ a.e.) such ...
1
vote
1answer
26 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
2
votes
2answers
56 views

When do evaluation and the integral sign “commute”?

This is a difficult question to put into words so it's much easier to write the math. Let $a$ and $b$ be given constants and $g(y) \equiv \int_a^b f(x,y) dx$. When is $g(c) = \int_a^b f(x,c) dx$? I ...
1
vote
0answers
15 views

the continuity of total variation function of a continuous function of bounded variation [duplicate]

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
2
votes
2answers
31 views

Establish the absolute maximum of a function

We have this function:$$f(x)=\begin{cases} \sin(x) \cdot\ln(\sin2x), & \mbox{if }0<x<\pi/2 \\ 0, & \mbox{if }x=0,\mbox{or }x=\pi/2 \end{cases}$$ So, how to prove that it decreases and ...
2
votes
1answer
35 views

holomorphic function with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
1
vote
3answers
52 views

Is Inverse Function Composition Commutative?

Given $f: \mathbb{R}\to(-1,1)$ is there a theorem that states $f\left(f^{-1}(x)\right) = f^{-1}\left(f(x)\right)$. In example, is $\tanh{\left(\tanh^{-1}{(x^2)}\right)} ...
3
votes
2answers
47 views

Composition of functions is constant in $\mathbb{R^2}$.

Let $\hspace{0.05cm}f:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ and $\hspace{0.05cm}$$g:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ be such that$\hspace{0.05cm}$ $g\circ f$$\hspace{0.05cm}$ ...
1
vote
0answers
29 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
0
votes
1answer
31 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
2
votes
1answer
52 views

Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m $ is injective then $n \leq m$?

I had intended to restrict the image then $f:U \rightarrow f(U) \subset \mathbb{R}^m $ is bijective. Therefore $\dim f(U) = n \leq m$. That's right?
0
votes
2answers
46 views

Period of $\frac{\sin(Ny)}{sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
4
votes
2answers
72 views

$f'$ is bounded and isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\lim_{x\to y}f'$ does not exist

Prove/disprove: $f$ has a bounded derivative and $f'$ isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\displaystyle\lim_{x\to y}f'$ does not exist. I think that if $f'$ ...
0
votes
1answer
33 views

Help me how to find the limit. In this case something is strange.

A fn(x) is a sequence of function. fn:[0,1]→R defined by fn(x) = n(1-x) × x^n. And I want to know limit of this function as n→∞. In my opinion we can see this like this n(1-x) × x^n. So as n→∞, ...
2
votes
1answer
37 views

right derivative of a continuous function

Let $f:(a,b)\longrightarrow \mathbb{R}$ be continuous. Suppose $D_+f(x)=\lim_{h\to 0+}\frac{f(x+h)-f(x)}{h}\geq 0$ for any $x\in (a,b)$. Prove that $f(x_1)\geq f(x_0)$ whenever $x_1\geq x_0$. How to ...
0
votes
2answers
41 views

construct functions such that $f(x)g(x)\gt0 $ and

Does there exist real functions $f, g\in C^1[-1,1]$ such that $$\det\left(\begin{array}{cc}f &g \\ f'&g'\end{array}\right)\equiv0 \qquad \det\left(\begin{array}{cc}\int_{-1}^1f^2\,\mathrm ...
0
votes
0answers
25 views

double limits and liminf, limsup

A theorem in calculus: Let $f:\mathbb{R}^2\longrightarrow \mathbb{R}$. If $\lim_{(x,y)\to (a,b)}f(x,y)=A$ exists (either finite or $\infty$) and there exists $\epsilon>0$ such that for any ...
1
vote
1answer
40 views

continuous extension and smooth extension of a function

Let $X$ be a metric space. Let $E$ be a subset of $X$. (1). any continuous function $f:E\longrightarrow \mathbb{R}$ can be extended to a continuous function $g: X\longrightarrow \mathbb{R}$ such ...
1
vote
1answer
35 views

Problems with a series and a wrong problem statement suspected

I've been fighting with the following task: The series $\displaystyle \sum_{n=1}^\infty f_n(x)$ is convergent in the points $a$ and $b$, and the functions $f_n(x)$, $n \in \mathbb{N}$, are monotonic ...
2
votes
2answers
30 views

uniformly convergence on compact metric space

Let $K$ be a compact metric space. Let $\{f_n\}_{n=1}^\infty$ be a sequence of continuous functions on $K$ such that $f_n$ converges to a function $f$ pointwise on $K$. on Walt. Rudin's book ...
1
vote
1answer
33 views

Riemann integration and the Fundamental Theorem of Calculus

Hello, I am unsure of how to define the partition to find a suitable result for the Riemann integrable, how do I do this for a discontinuous function?
1
vote
1answer
42 views

find the maximum of the function F under the condition $ \sum_{i=1}^N x_i = 1$

Let F a function of $ \mathbb{R} ^N_+ \rightarrow \mathbb{R}$ defined as : $$F(x_1,..,x_N)= - \sum_{i=1}^N x_i log(x_i) , x_i \gt 0$$ How can i find the maximum of the function F under the ...
0
votes
3answers
72 views

Can two function $f$ and $g$ have same values through out a given interval and different values outside that interval?

Is it possible that for two functions $f$ and $g$ and some interval $(a,b)$ we have $f(x)=g(x)$ for all $x\in(a,b)$ and $f(x)\neq g(x)$ for $x$ outside the interval $(a,b)$? $f$ and $g$ are ...
1
vote
1answer
48 views

Differentiability of a convex function

Let $f,g\colon \mathbb{R}\rightarrow \mathbb{R}$ be convex functions such that $f\ge g$ and $f(0)=g(0)$. Show that if $f$ is differentiable in 0, then $g$ is too and $$ f'(0)=g'(0)$$ I have no idea ...
0
votes
1answer
19 views

Finding a function without knowing its structure but some conditions

I'm trying to find a function who meets this conditions but have no idea where to start. Just think it may be related to the function $Ca^{-\left(x-\mu\right)^2}$, If it really has this structure (or ...
1
vote
1answer
114 views

Stacked with this Problem of Calculus

I have been struggling for quite some time with the following problem and I would really appreciate some help: Consider ...
1
vote
1answer
102 views

Harmonic series

Is there a sequence that converges to zero such that the series over the product of every summand of the harmonic series with the appropriate element of the sequence is not convergent?
2
votes
3answers
205 views

Limit of $\displaystyle \frac{1-\sin x}{1+\sin x}$

I have to find the limit of $\displaystyle \frac{1-\sin x}{1+\sin x}$ as $x\to\infty$. I was studying this function finding its real graph but to do this I need to know where the function goes as $x$ ...
5
votes
2answers
124 views

Why is $f'(x) < f(x)/x$ for $f''(x)<0$

How can it be shown formally that for all $x>0$ $f(x) > 0$, $f'(x) > 0$, and $f''(x) < 0$ imply $f'(x) < \frac{f(x)}{x}$? My (somewhat sloppy) intuition is this: since $f'(x)$ is ...
1
vote
0answers
47 views

Maximal value of an infinite set. [duplicate]

Consider a continuous function $f$ over an interval $[a,b]$. Let $S$ be the set of all values that $f(x)$ takes over $I$. Intuitively speaking, I believe this set has a maximal and minimal value. Is ...
1
vote
1answer
112 views

When to rationalize numerator and/or denominator?

Sometimes, we have to rationalize either the numerator or the denominator, and sometimes we can still work the problem without rationalizing. So, in some cases, rationalizing can be done, although it ...
0
votes
2answers
53 views

Real function injectivity proof

How can one prove that the real function $f(x) = (1-x)x^{\frac{x}{1-x}}$ is an injection?
1
vote
3answers
107 views

A non-constant, increasing function $f$ such that $f(b)=\int_a^bf$

Is there a non-constant, increasing function $f\colon A\to B$, where $A,B\subset\mathbf{R}$ such that $$f(b)=\int_a^bf(x)\;\mathrm{d}x$$ for $a,b\in{A}$ with $a<b$.
5
votes
0answers
54 views

Is there a bijection $f:\mathbb R^+\to\mathbb R^+$ s.t. $f'(x)=f^{-1}(x)$? [duplicate]

Is there a bijection $f:\mathbb R^+\to\mathbb R^+$ s.t. $f'(x)=f^{-1}(x)$? And if there is, can I prove uniqueness? This problem troubles me.
1
vote
2answers
302 views

Finding all the points (x,y) where $f(x,y)=\sqrt{|x^3y|}$ is differentiable.

Finding all the points (x,y) where $f(x,y)=\sqrt{|x^3y|}$ is differentiable. I started off with this: If a function is differentiable then: $f(x_0+\Delta x,y_0+\Delta y)=f(x_0,y_0)+\frac {\partial ...
0
votes
3answers
97 views

Proof: function is discontinuous (via Limit-Criterion!)

I'd like to show that the function, $$f(x) = \begin{cases} 0 & x \not\in \mathbb{Q}\\ 1 & x \in \mathbb{Q} \end{cases}$$ is discontinuous via using the sequence-limit-criterion: And I'd ...
4
votes
3answers
2k views

Proof for Dirichlet Function and discontinuous

I think I don't understand how it works.. I found some proofs.. okay, let's see: Well I'd like to show that the function, $$f(x) = \begin{cases} 0 & x \not\in \mathbb{Q}\\ 1 & x \in ...
3
votes
4answers
119 views

A smooth function instead of a piecewise function

I want to find a smooth function approximating f(x) as best as possible: \begin{equation*} f(x) = \begin{cases} x & \text{if } x \le a,\\ a & \text{if } x > a. \end{cases} \end{equation*} ...
2
votes
1answer
63 views

Is constructing a function that DNE a sufficient counterexample to show the function does not diverge to $\infty$?

Prove or disprove: If $f(x)\to 0$ as $x\to a^+$ and $g(x)\geq 1$ for all $x\in \mathbb{R}$, then $g(x)/f(x)\to\infty$ as $x\to a^+$. Counterexample: Let $f(x)=0$ and $g(x)=1$ for all ...
5
votes
1answer
625 views

Prove or disprove: if $f$ and $fg$ are continuous then $g$ is continuous.

Prove of provide a counterexample: Suppose that $f$ and $g$ are defined and finite valued on an open interval $I$ which contains $a$, that $f$ is continuous at $a$, and that $f(a)\neq 0$. Then $g$ is ...
30
votes
4answers
1k 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
468 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
125 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
121 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
291 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
502 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
2k 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 ...
17
votes
3answers
667 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
203 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
117 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 ...