0
votes
0answers
16 views

Unbounded Function

I am trying to find vales of $a >0$ s.t the function is unbounded on $[0,1]$ $f_a(x)= \begin{cases} x^{a-2}(ax\sin(1/x)-\cos(1/x)), & x\neq 0 \\ 0, & x =0 \end{cases}$
0
votes
0answers
9 views

composition of Riemann integrable functions.

I have two functions: f:[a,b]->R and g:[c,d]->R where a My question is if it follows that g o f (the composite function of f,g) is Riemann integrable as well?
1
vote
1answer
34 views

Is there a basis for the continuous functions space?

I've been searching all over the Internet for this but without finding a satisfying answer. This might be a dumb question, but I would like to know the answer anyway. Is there a set of continuous ...
4
votes
1answer
146 views

Is the range of an injective function dense somewhere?

Consider an injective function $\,\,f:[0,1]\rightarrow[0,1].\,$ Then is it true that there always exists some non-empty open subinterval of $[0,1]$, such that $f([0,1])$ is dense in that subinterval? ...
1
vote
1answer
31 views

For $f(x, y) = x-y$, is $f(K \times K)$ closed if $K$ is closed?

$f: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x, y) = x-y$. For $K \subset \mathbb{R}$ closed is $f(K\times K)$ closed? For the closed interval this is straight forwardly true ...
2
votes
2answers
116 views

Is the function $\,f(x, y) = x-y\,$ closed?

Is $f: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$, such that $f(x,y)=x-y$ closed?
1
vote
3answers
71 views

Prove this limit (formally)

As I was coursing through Spivak's calculus, more like analysis; I found an interesting, questionable example. Let $\frac{p}{q}$ be in its lowest terms; $p$ and $q$ are integers with no common ...
0
votes
0answers
52 views

Connection between the Cantor set and the Tent map $T_3$

I would like to prove part c) using the ternary expansion as shown in the second half of the image. I understand how to compute up to what is shown in the image. I am struggling to understand the ...
1
vote
1answer
34 views

Continuous functions unbounded on set

For Each of the sets construct a continuous function that is unbounded on the set. $\Bbb N$ $(2,3)$ $\left\{\frac 1 n \mid n \in \Bbb N\right\}$ $[0, \sqrt 2]\cap \Bbb Q$ ...
0
votes
0answers
37 views

If f is a continuous function on $[0,1]$ and if for each $n=0,1,2,3,…$. Prove that $f=0$ on $[0,1]$.

If f is a continuous function on $[0,1]$ and if for each $n=0,1,2,3,...$ $$ \int_0^1 f(t)t^n dt=0$$ Then prove that $f=0$ on $[0,1]$. Here I don't want the proof. I have one proof. But I have ...
2
votes
0answers
44 views

Sign of a Function over $[0,\pi]$

Let $\lambda \in \mathbb{R} $ such that $\left| \lambda \right| \ne 1$ $$f_\lambda (x)=(λ\cos x−1)(λ−\cos x)$$ Show that : For $|λ|<1$, $$f_λ(x)=\begin{cases} <0 ...
0
votes
1answer
13 views

How to alter the interval of a composite function

Let $f, g : R → R$ $$f(x) = \begin{cases} x + 3 &\text{if } x ≥ 0,\\ x^2 &\text{if } x < 0 \end{cases}$$ $$g(x) = \begin{cases} 2x + 1 &\text{if } x ≥ 3,\\ x ...
1
vote
1answer
70 views

Prove the following function is differentiable

I have to prove if this function is differentiable. $$f(x,y)= \begin{cases} (x^2+y^2) \sin\frac 1{(x^2+y^2)} \iff (x,y) \neq (0,0) \\0 \iff (x,y)=(0,0) \end{cases}$$ I tried proving that all of its ...
2
votes
4answers
278 views

Is the function continuous

$$f(x)=\begin{cases} x^{2}-1 & x \leq 1\\x- \frac{1}{x} & x \geq 1\end{cases} $$ How can you show if these kind of functions are continuous or not?
2
votes
1answer
143 views

Find $f$, such that $\,f,f',\dots,f^{(n-1)}\,$ linearly independent and $\,f^{(n)}=f$

I am trying to find a function $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{C})$, satisfying the differential equation $$ f^{(n)}=f, $$ and with $\,f,f',\dots,f^{(n-1)}\,$ being linearly independent. ...
1
vote
1answer
36 views

A Real valued function which is discontinuous **only** on a given specific set.

Let $\mathbb{L}=\{x_n \ |\ n=1,2,3 \dots\}$ be a countable subset of $\mathbb{R}$. My aim is to construct a real valued function $f$ on $\mathbb{R}$ such that $f$ is discontinuous at every point ...
0
votes
1answer
15 views

Period of two equal functions

I'm dealing with a problem here. We know that two functions are the same if they have the same domain and codomain. Let's say we have given the functions $f$ ang $g$ where ...
0
votes
1answer
42 views

Show that R is an equivalence relation on X for x, y in X iff f(x) = f(y)

$f:X→Y$ $x,y ∈ X,xRy$ iff $f(x) = f(y)$ Show that R is an equivalence relation on X. Also when $X = Y = \mathbb{R}$ and $f: \mathbb{R} \to \mathbb{R} $ with $x \mapsto x^2$ for all $x∈R$ find the ...
1
vote
1answer
29 views

A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$

A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$ I am trying to prove the triangle inequality of this norm. So far I have that: \begin{align} ...
2
votes
0answers
34 views

rewriting the inverse image

If $\phi_k:\mathbb{R}^2\rightarrow \mathbb{R}$ are continuous functions, for all $k\geq0$ and $$\phi=\limsup_{n\rightarrow \infty }\phi_n$$ Let $A\subset \mathbb{R}$, is possible to write ...
0
votes
2answers
68 views

Existence of a root

Let $f:[a,b] \rightarrow \Bbb R$ continuous, such that for every $x$ there is a $y$ such as that $|f(y)|\leq|f(x)|/2$. Show there exists a $\xi$ such that $f(\xi)=0$
0
votes
1answer
46 views

Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...
1
vote
0answers
26 views

Rules to evaluate Suprema and Infima

Note: All functions considered are supposed to be bounded. Nowhere I found rules to evaluate suprema and infima. Obviously, $$ c \cdot \sup_{x\in \mathbb R} f(x) = \sup_{x\in\mathbb R} cf(x) ...
-1
votes
2answers
38 views

Separating a Complex Valued Function

Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $ f(z)=u(x,y) + iv(x,y)$? Thank You, C.A
0
votes
2answers
83 views

Real and imaginary parts of a complex-valued function

How do you get a complex-valued function $ f(z) = f(x+iy) = \frac{z^{s-1}}{e^{-z}-1}, $ where $s$ is a constant complex number and $z$ is a complex variable, into the form: $ f(x+iy) = a(x,y) + ...
0
votes
1answer
19 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
31 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
58 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
45 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 $f(D)$?
0
votes
0answers
50 views

How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
1
vote
3answers
69 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
59 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
1answer
35 views

Having derivative in some $x_{0}$ implies having it in $U(x_{0})$ [closed]

Let $f$ be continious function in R and it has a derivative in $x_{0}$. Does it have derivative in some $U(x_{0})$?
0
votes
1answer
128 views

if $\mid f(x)\mid<1$ then $\mid f'(x)\mid<1$

I am wondering, if $\mid f(x)\mid<1$ then $\mid f'(x)\mid<1$. Is this true? I can not find counter example.
1
vote
0answers
30 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
33 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 ...
1
vote
1answer
39 views

Find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$

How can I find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$. I've tried derivating it but didn't reach any result.
0
votes
0answers
42 views

Generalizations of functions

I'm trying to collection examples of mathematical entities that are generalizations of functions. The use of the word "generalization" here doesn't need to be strict, as in every function is an $X$ ...
1
vote
2answers
111 views

Finding a root of a function via Rolle's theorem

Consider the function $f(t)=a(1-t)\cos(at)-\sin (at)$, where $a\in\mathbb R$. To show that it has a root in the unit interval I am urged to integrate $f$ and apply Rolle's Theorem. Attempt: $$\int ...
1
vote
2answers
65 views

If a uniquness for all functions exist shouldn't there be uniquness to recursion?

What I'm specifically saying is every function is definitely unique, as they may be nearly equivalent to another function, for example. Let's make a table of values for $^{x}2$ (0,1) (1,2) (2,4) ...
2
votes
1answer
62 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?
3
votes
1answer
36 views

Election measurable in uniform continuity

Let $f:[0,1]\times [0,1] \rightarrow \mathbb{R}$ borel measurable such that for all $x \in [0,1]$ $f(x,-):[0,1] \rightarrow \mathbb{R}$ is continuous, in particular uniformly continuous. Then there ...
2
votes
1answer
87 views

Show that $e^{-a|x|}$ does not belong to Schwartz space

Let $f : \mathbb R \to \mathbb R$ and $a > 0$ given by $f(x) = e^{-a|x|}$. Show that $f$ is rapidly decreasing and belongs to $L_1(\mathbb R)$, but not to $\mathcal S(\mathbb R)$. I had shown that ...
0
votes
2answers
47 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
78 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
2answers
67 views

Example of a function?

$f$ is a discontinuous and bounded function defined on a closed set $C$. Also there exists a non-discrete closed subset in the image of $f$ such that it's inverse is open. Can you give an example ...
2
votes
0answers
45 views

Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
0
votes
1answer
44 views

Interesting question regarding elementary functions

I had this question at a test for a job interview and since and I didn't solve it. Some time later i still can't figure it out, so any insight is helpful. You need to write a function $f(x)$ such ...