Tagged Questions

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Basic calculus question with continuous function [duplicate]

This is actually not my question, it was asked yesterday by user176744 in this link $[0,n]$ continuous function problem and I feel as if it didn't get enough attention. I am also interested in a ...
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A function such that $f'(x)>0$, but not strictly monotone increasing.

This is Exercise 10.3.5 from Analysis Vol.1 by Terence Tao. Give an example of a subset $X \subset \mathbb{R}$ and a function $f: X \to \mathbb{R}$ which is differentiable on $X$, is such that ...
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is this a valid counter-example - function is not locally invertible

Let $S_n$ be the set of all symmetric matrices with real entries of size $n$x$n$. We are asked if the function $f:S_n \to S_n$, $f(A)=A^2$ is locally invertible for every $A$ (Using the Inverse ...
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$\mathcal{C}^\infty$ 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 ...
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L'Hospital's rule vs Taylor series

One classical application of Taylor expansions is to obtain polynomial equivalents of complicated functions and use them to compute limits. For example, with Landau notations, we have ...
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A counterexample

Let $f:\mathbb{R}\to(0,\infty)$ a locally integrable function. I want to compare these two conditions $$\limsup_{r\to + \infty}\frac{r}{\int_{-r}^r f(x)dx}<+\infty. \tag{1}\label{1}$$ and ...
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smooth function whose (n+1)th derivative is defined only on a propersubset of the domain of the nth, and the radius contract to 0

So I'm basically wondering if there exists such a function, whose (n+1)th derivative is defined only on a proper subset of the domain where the nth derivative is defined, and with the property that ...
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A real continuous periodic function with two incommensurate periods is constant.

I think I have a proof for the statement, but I can't think of a counter-example when $f: \mathbb{R} \to \mathbb{R}$ is not continous. Here's the problem: Let $f: \mathbb{R} \to \mathbb{R}$ be a ...
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Is a bounded continuous function defined on $\Bbb R$ differentiable?

Is a bounded continuous function defined on $\Bbb R$ differentiable? Why so? The query is fueled by the following question: Let $f : \Bbb R \rightarrow \Bbb R$ be a bounded continuous function. ...
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If $\lim \limits_{x \to \infty}f(x) = L \neq 0$ must it be that $\lim_{x \to \infty} f(x) \sin x$ does not exist?

I got this question: Let $f\colon \mathbb{R} \to \mathbb{R}$ be a function that satisfies $\lim \limits_{x \to \infty}f(x) = L \neq 0$, Must it be that $\lim_{x \to \infty} f(x) sin x$ does not ...
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Are continuous functions monotonic for very small ranges?

So I am wondering, if we have a continuous function f and we take the range $[c,c+h]$ for $h \to 0$, is the function monotonic in that range?
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Is a $C^\infty$ function local monotone at a minimum?

The following is a modified exercise from an analysis 1 book. Is there a function $f:\mathbb{R} \rightarrow \mathbb{R}$ with: i) $f$ has in $0$ a strict local and global minimum. ii) $f\in C^\infty$ ...
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Existence of an infinitely differentiable function $f$ with ${f^{(n)}}(0) = 0$ for all $n \in \mathbb{N}$.

How can one show that there exists an infinitely differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that ${f^{(n)}}(0) = 0$ but $f^{(n)} \not\equiv 0$ for all $n \in \mathbb{N}$?
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If $|f(x)|$ is a differentiable function, then $f(x)$ is also?

If $|f(x)|$ is a differentiable function, then $f(x)$ is also a differentiable function. Why is this wrong? Can you find a counterexample please? It seems like a true sentence.
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Non-physical Jounce Examples in Nature

What are some good examples of jounce, the fourth derivative of position, in the non-physics arena? The reason I ask is that A) it's already difficult for a lay to visualize it in the physical arena ...
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nonstandard example of smooth function which fails to be analytic on $\mathbb{R}$

When I teach second-semester calculus I usually discuss the function $f$ defined by $$f(x)=e^{-1/x^2}$$ for $x \neq 0$ and $f(0)=0$. Or, almost the same example, $g$ defined by $$g(x)=e^{-1/x^2}$$ ...
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Is there any complex-valued $C^\infty$ function $f(z)$ coincide with $z^{3/2}$ infinitely many times?

I want a $C^\infty$ function $f:U\rightarrow\mathbb{C}$, where $U\subset\mathbb{C}$ is a neighborhood of $0$, such that there exists a sequence of points $z_n\in U-\{0\}$ and ...
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Sequence convergence and parentheses insertion

find an example for a series $a_{n}$ that satisfies the following: $a_{n}\xrightarrow[n\to\infty]{}0$ ${\displaystyle \sum_{n=1}^{\infty}a_{n}}$ does not converges There is a way to insert ...
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Set of zeroes of the derivative of a pathological function

For a continuous function $f : [0,1] \to {\mathbb R}$, let us set $$X_f=\lbrace x \in [0,1] \bigg| f'(x)=0 \rbrace$$ (for a $x\not\in X_f$, $f'(x)$ may be a nonzero value or undefined). There ...
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Real-valued function of one variable which is continuous on [a,b] and semi-differentiable on [a,b)?

Is there any real-valued function of one variable which is continuous on [a,b] and right differentiable on [a,b), but not left differentiable at any point?
Construct an example of a differentiable function such that $$\forall r \in {\Bbb Q}\quad f(r) \in {\Bbb Q}\text{ but } f'(r) \notin {\Bbb Q}$$ this example is not trivial, in a paper they ...
I tried to prove something but I could not, I don't know if it's true or not, but I did not found a counterexample. Let $(a_n)$ be a sequence in a general metric space such that for any fixed \$k ...