# Tagged Questions

Elementary questions about functions, notation, properties, and operations such as function composition.

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### K-Map reduction

There's an exercise which states that depending on certain rules a led(of different colour) shall turn on or not. There are four leds, so I've made four functions (One each led, through Karnaugh Map ...
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### 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 ...
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### Is $C(C(\mathbb R))$ notation for the set of continuous functions mapping $C(\mathbb R)$ to itself?

Given that in general functional analysis we have $C(\mathbb{R})$ being the set of all continuous functions, $f: \mathbb{R} \to \mathbb{R}$. However, could I use $C(C(\mathbb{R}))$ notationally to be ...
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### Properties inherited by $f\circ g$ from $f$

Suppose $f,g:\mathbb{R}\to \mathbb{R}$ Prop: Suppose $g$ and $f \circ g$ are ______, and $g$ achieves every value in $\mathbb{R}$. Then $f$ is ______. If in the blanks we put the word ...
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### The meaning of product of functions in multivariable calculus

If $f$ and $g$ are $2$ functions $\mathbb{R}^n\rightarrow\mathbb{R}^m$ For $m=1$ and $n>1$ is $f\cdot g$ or $(f\cdot g)(x)$ defined for? Would that be a real number? And for $n=1$ and $m>1$, is ...
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### does a positive/negative number cancel itself?

By positive negative I mean the function that looks like an addition sign $(+)$ with a subtraction sign $(-)$ right underneath it. does a positive/negative number cancel itself? As in $x$ ...
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### Successive Differentiation of $\mathrm{e}^{g(t)}$

I am trying to find the closed for solution for $A_n$. Assume $A_0 = g'(t)$, $A_1 = g'(t)$, and $$\dfrac{d^n}{dt^n}\left[e^{g(t)}\right] = A_n e^{g(t)}$$ The problem has a recursive relationship of ...
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### Prove there's $M>0$ such that: $f(x)\le Mx^2$

Let $f:[-1,1]\to\mathbb{R}$, three-times differentiable function and $f(0)=0$, $f(x)\ge 0$ for all $x\in[-1,1]$. Prove there's $M>0$ such that $f(x)\le Mx^2$. Hint: use Taylor formula. So ...
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### Prove $f$ isn't uniformly continuous

I already proved (followed by an hint) that $f(y)-f(x) > x(y-x)$ for all $y>x>0$. I need to prove $f$ isn't uniformly continuous on $(0, \infty)$. What I did: Lets assume by contradiction ...
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### Expressing the area as a function :)

Express the area A of an equilateral triangle as a function of the height of the triangle. Thanks :) I am not sure where to even start on how to answer this problem.
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### $\lim_{x \to \infty} f(x)=1$ $\implies$ $f(x) \sin x$ is uniformly continuous on $\mathbb R$?

Let $f:\mathbb R \to \mathbb R$ be a continuos function such that $\lim_{x \to \infty} f(x)=1$ , then is it true that $f(x) \sin x$ is uniformly continuous on $\mathbb R$ ?
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### Intersection of Images of a function

I'm trying to understand intuitively why the image ( under some function ) of the intersection of subsets of the domain of that function is only contained ( and not equal ) to the intersection of the ...
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### Why doesn't this converge?

Why doesn't $$\int_{-1}^1 \frac{1}{x}~\mathrm{d}x$$ converge? I mean you would think that because of symmetry the area from the negative side and positive side cancel out, resulting in the integral ...