All Questions
Tagged with functions solution-verification
1,052
questions
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30
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Proving there exist $g,h$ where $g = \Theta(h)$ and $f(x) = g(x) - h(x)$ for a function $f$
I am trying to prove that for any function $f : \mathbb{Z}^{+}\rightarrow \mathbb{R}^{+}$, there exist $2$ functions $g : \mathbb{Z}^{+}\rightarrow \mathbb{R}^{+}$ and $h : \mathbb{Z}^{+}\rightarrow \...
- 2,947
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votes
1
answer
49
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Proving $f + c = O(f)$ doesn't always hold- where is my mistake?
I seem to have proved the following statement false: that for any function $f : \mathbb{Z}^{+}\rightarrow \mathbb{R}^{+}$ and any $c \in \mathbb{R}, f +c = O(f)$, where for any $2$ functions $f : \...
- 2,947
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votes
2
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62
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Condition for a function to change its sign at $x=a$
Consider a situation in which we are only allowed to evaluate the value of a real function $f$ (and its derivatives) at a particular value $a$. Also, assume that the given function is $C^\infty$ ...
- 1,392
2
votes
1
answer
61
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Finding the different zeros of a continuous function
I'm working on Spivak's Calculus and am doing Problem 4 of Chapter 8. Here's the problem:
Suppose $f$ is continuous on $[a,b]$ and that $f(a) = f(b) = 0$. Suppose also that $f(x_0) > 0$ for some $...
- 249
2
votes
3
answers
65
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Proof that the supremum of a continuous function is part of the range of that function
I'm following the textbook "Calculus" By Spivak. Currently, I'm reading a proof given for the following theorem:
"If $f$ is continuous on $[a,b]$ then there exists an $x^*\in[a,b]$ such ...
- 249
0
votes
1
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30
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Under what hypotheses is the primitive function bijective?
I am trying to determine under what assumptions the function
$$F:(0,\infty) \to (0,\infty),$$
defined by
$$F(t) = \int_{0}^{t} f(s)ds$$
is a bijection. For injectivity, simply require that $f$ ...
- 3,002
2
votes
1
answer
30
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Conditions on the constants for two "linear" functions to commute under the operation of function composition
Let $a$, $b$, $c$, and $d$ be any complex numbers such that $a \neq 0$ and $c \neq 0$; let the functions $f \colon \mathbb{C} \longrightarrow \mathbb{C}$ and $g \colon \mathbb{C} \longrightarrow \...
- 26.9k
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3
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113
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Find $f(x)$ assuming that $f(\sin x)+f(\cos x)=2x-\frac{\pi}{2}$
If $f(x)$ is a real valued function such that $$f(\sin x)+f(\cos x)=2x-\frac{\pi}{2}$$
Find $f(x)$.
I did $x\to\arcsin x$ and then $x\to \arccos x$ and I obtained $2\arcsin x=2\arccos x$ or $x=\frac{...
1
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Tao Analysis I Exercise 3.5.12(i) on Recursively Defined Functions
I am a self-teaching beginner and need some advice on my attempt to do part (1) of Exercise 3.5.12 from Tao's Analysis I 4th ed.
My question is specifically about the following steps in the proof:
...
- 3,325
2
votes
1
answer
111
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Prove that if $f: U \rightarrow \mathbb{R}^n$, then $f=(f_1,...,f_n)$
I gave myself a claim to prove:
If $f: U \rightarrow \mathbb{R}^n$, then we can write $f$ as $f=(f_1,...,f_n)$.
I tried to prove it this way:
Assume $f: U \rightarrow \mathbb{R}^n$. Then we know that $...
- 45
2
votes
0
answers
33
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Different Formulations of Differentials as Generating Transformations: $e^{t A}f(x) = h(x,t); A=\frac{\partial_x}{g'(x)} \& h(x,t)=f(g^{-1}(g(x)+t))$
For context and introduction please see:
Perspective on Differentials Generating Transformations of Functions - with Fourier Transformations $e^{a\partial_x} f(x) = f(x+a)$
Here I used Fourier ...
- 147
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votes
1
answer
125
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If $f:\mathbb{R}-\{-1,K\}\rightarrow\mathbb{R}-\{\alpha,\beta\}$ is a function given by $f(x)=\dfrac{(2x-1)(2x^2-4 px+p^3)}{(x+1)(x^2-p^2x+p^2)}$
If $f:\mathbb{R}-\{-1,K\}\rightarrow\mathbb{R}-\{\alpha,\beta\}$ is a bijective function defined by $f(x)=\dfrac{(2x-1)(2x^2-4
px+p^3)}{(x+1)(x^2-p^2x+p^2)}$, where $p\geq 0$, then which if the ...
- 3,835
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1
answer
42
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Perspective on Differentials Generating Transformations of Functions - with Fourier Transformations $e^{a\partial_x} f(x) = f(x+a)$
An analytic function $f(x)$ can be transformed by the exponential of a differential operator.
The most known and easiest example is
$
e^{a \partial_x} f(x) = f(x+a)
$
Generally this is shown by Taylor ...
- 147
-1
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3
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71
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Let$f:R\to R$ be a continuous function such that for all $x\in R$ and for all $t\ge0\; f(x)=f(e^tx)$ show that $f(x)$ is a constant function.
Let $f:R\to R$ be a continuous function such that for all $x\in R$ and for all $t\ge0\; f(x)=f(e^tx)$ show that $f(x)$ is a constant function.
I do have a proof but am not satisfied enough if so want ...
- 868
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Clarification on function properties: injective and surjective
I want to see if I am on the right path for if we have a $f: \mathscr P( \Bbb N) \to \Bbb N \cup \{ \infty \}$ be the function which assigns to each subset $A \subseteq \Bbb N$ the cardinality of that ...
- 147