# Tagged Questions

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### Convergent Sequence of Analytic Functions, Do Their Derivatives Converge?

If you have a sequence of real analytic functions that converge on every compact subset in your domain, do their derivatives necessarily converge to the derivatives of the function that they converge ...
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### Epsilon-Delta continuity definition for straight lines parallel to axes

I am taking a course on real analysis online and I encountered the $\epsilon-\delta$ definition for a function to be continuous. But I wonder if I can apply it to functions which are straight lines ...
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### Functional characterization of zeroth law of thermodynamics [Sepration of Variables]

Zeroth law of thermodynamics is stated also as: If A is in thermal equilibrium with B and if B is in thermal equilibrium with C, then A is in thermal equilibrium with C. This can be formulated ...
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### How $n^d \times m([0, \frac{1}{n}[^d) = m([0, 1[^d)$ follows from translation invariance and (finite) additivity

In this StackExchange question (which itself seems to reference to an exercise in Terence Tao's lecture notes on introductory measure theory on his blog here), it's said that assuming "finite ...
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### To prove : If $f^n$ has a unique fixed point $b$ then $f(b)=b$

If $f: \mathbb R \to \mathbb R$ be a function such that for some $n_o \in \mathbb N$ , the $n_o$th iterate of $f$ has a unique fixed point $b$ , then how to prove that $f(b)=b$ ? I cant think of ...
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### Existence of function satisfying given conditions?

Let $f:[0,1]\longrightarrow[0,1]$ be continuous, strictly increasing and $f(1)=1$. Suppose further that $f(x)>x$ for all $x\in[0,1)$. Is there any function satisfying the above conditions? My ...
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### Search for two Real Valued functions.

Can we have two real valued functions $f_1$ and $f_2$ defined on $[a,b]$ such that $f_1(x)=f_2(x)$ for infinitely many points and $f_1(x)\neq f_2(x)$ for infinitely many points. ?
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### Can all functions be expressed in terms of elementary functions?

After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me: Can the non-elementary functions be decomposed to ...
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### Derive property from continuity - is this proof valid?

Prove that if $f:R^+ \rightarrow R^+$ is continuous on the positive reals and is decreasing, then for all $a$ there exists an $\eta > 0$ such that $(a-\eta)f(a-\eta) > \frac{1}{2}a*f(a)$. EDIT ...
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### Function such that zeros$=$order of the derivative

Does there exist a function $f\in C^n(\mathbb{R},\mathbb{R})$ for $n\ge2$ such that $f^{(n)}$ has exactly $n$ zeros, $f^{(n-1)}$ has exactly $n-1$ zeros and so on ? Where $f^{(n)}$ is the nth ...
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### Example of a smooth 'step'-function that is constant below 0 and constant above 1

I need an infinitely smooth non-decreasing function $\ f(x)$, that $$f(x)=0\quad\forall x\leq 0,$$ $$f(x)=1\quad\forall x\geq 1,$$ and all its derivatives in $x=0$ and $x=1$ are $0$. I found that I ...
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### If $f$ is continuous, so is $g=|f|$ [closed]

Prove that if $f$ is continuous, so is $g=|f|$. I need help on this. Thank you. Ok, this is my first time here. The definition of continuity i am using is that $f$ is continuous at $a$ if for any ...
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### Covering $\mathbb R^2$ with function graphs

Suppose we have a countable family of function graphs (each function is $\mathbb R\to\mathbb R$, not necessary continuous). Obviously, they cannot cover the whole plane $\mathbb R^2$, because they ...
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### Prove the existence of exactly two maxima for a positive $L^1$ function

I have a function $f:\mathbb R \rightarrow \mathbb R^+$ with the following properties it is $L^1$ and $C^2$ it has one single extremum (maximum) at $x=0$ it is symmetric: $f(x)=f(-x)$. it is ...
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### Behavior of Two Functions

If two functions can be shown to agree at an infinite number of points, what additional information would be required to show that these two functions are equivalent? For example, if two polynomials ...
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### The function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$.

Show that if $f\in \mathcal{C}^3$ and $2\cdot\pi$ periodic then the function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$. My attempt : f is $2\pi$ periodic and $\mathcal{C}^3$, we have : ...
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### Composition of functions is constant.

If $f\circ g$ ($f$ composed with $g$) is constant, then which of the following is constant? $f$ $g$ $g \circ f$ ($g$ composed with $f$) Both $f$ and $g$ Both $g$ and $g \circ f$ ...
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### Functions that are continuous only at two points?

I need to find a function $f:\mathbb{R}\to\mathbb{R}$ which is continuous only at two points, but discontinuous everywhere else. How on earth would I go about doing this? I can't think of any ...
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### Show that there exist $g\in C^{\infty}$ such that $f(x)=g(x^2)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$, even,. Show that there exist $g\in C^{\infty}(\mathbb{R^+},\mathbb{R})$ such that for all $x\in \mathbb{R}, f(x)=g(x^2)$ My attempt : Let $g$ ...
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### Show that $f$ is bounded at $(0,+\infty)$.

Show that a function $f\in C^{1}\bigl((0,+\infty)\bigr)$ which satisfy $$f'(x)=\frac{1}{1+x^{4}+\cos f(x)},\, x>0$$ is bounded at $(0,+\infty)$. My attempt : I would like to prove that ...
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### Additive like function representation [duplicate]

Let's $f:R\rightarrow R,~r\gt0.$ And we have that for every $x,y\in R\Rightarrow |f(x+y)-f(x)-f(y)|\le r.$ Prove that there are $h:R \rightarrow R$ additive and $g:R \rightarrow [-r,r]$ functions such ...
### $\mathcal{C}^\infty$ strictly monotone function $\lim f(x) = 0$, but $\lim f^\prime(x) \ne 0$
Does there exist a strictly monotone function $f\colon \Bbb R\to\Bbb R$ that is $\mathcal{C}^\infty$, and $\lim\limits_{x\to+\infty} f(x) = 0$, but $$\lim_{x\to+\infty} f^\prime(x)$$ and ...
### Finding the maximum value of $\displaystyle \frac{x^n-a\sin x}{x^{n+1}}\ (x\gt 0)$
For $a\ge 1\in\mathbb R,n\ge 1\in\mathbb N$, let us define $f(x)$ as $$f(x)=\frac{x^n-a\sin x}{x^{n+1}}\ \ \ (x\gt 0).$$ Also, let $M(a,n)$ be the maximum value of $f(x).$ We may be able to know the ...