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17h
comment Notation for compactly supported functions
@KeenanKidwell Yes, with remarkable exceptions. For instance, Reed and Simon use $C_0^\infty$.
1d
comment Notation for compactly supported functions
@KeenanKidwell The notation $C_0^\infty(\mathbb{R}^N)$ to denote the set of test functions $\mathcal{D}(\mathbb{R}^N)$ is not so rare. I agree that it is not consistent with the notation for sequences, where $c_{00}$ is the set of sequences with compact support.
1d
comment Density of spaces $C_0^{\infty}(\mathbb{R})$, $W_2^2(\mathbb{R})$ and $L^2(\mathbb{R})$ in each other
@Dennis Welcome to the world of nonlinear analysis! You never find the exact statement that you need ;-) Actually, the density of compactly supported functions is proved in both books for the space $W^{1,p}$. The general case is true although it is not precisely stated as a theorem. Another possible source is Giovanni Leoni's book on Sobolev spaces, published by the American Mathematical Society.
1d
asked Notation for compactly supported functions
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answered Density of spaces $C_0^{\infty}(\mathbb{R})$, $W_2^2(\mathbb{R})$ and $L^2(\mathbb{R})$ in each other
1d
revised cauchy sequence in metric space
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comment Absolute Value Inequality Proof
What you are asking is really unclear to me.
1d
revised Conditions for convergence of derivatives from pointwise convergence
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answered Conditions for convergence of derivatives from pointwise convergence
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comment Conditions for convergence of derivatives from pointwise convergence
Two functions that are uniformly close may have rather different derivatives...
1d
comment Does this theorem concerning upper and lower bnound of a monotone decreasing function have a formal name?
I may be missing something, but it seems to me a very easy exercise that everybody does when studying the Riemann integral for the first time. For sure there exists a textbook in the world where it is proved, but I suspect that you can prove it by yourself.
1d
answered Is the total differential the same as the directional derivative?
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comment Does this theorem concerning upper and lower bnound of a monotone decreasing function have a formal name?
Can you please add a link to the blog?
1d
answered $A \subseteq B$ if and only if $B^c \subseteq A^c$
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answered Equivalent definition of limit of a function (Reference request)
2d
comment If $f\in C^2(\mathbb R)$ then $M_1^2 \le 2M_0 M_2$, where $M_k = \text {sup}_x |(d/dx)^k f(x)|$ for $k=0,1,2.$
This is a case of the so-called Laundau-Kolmogorov inequalitites: en.wikipedia.org/wiki/Landau%E2%80%93Kolmogorov_inequality
2d
comment Differentiable not $C^1$ and Darboux property
Look here: en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)
2d
comment Differentiable not $C^1$ and Darboux property
I do not understand: any differentiable function has the Darboux property. Can you explain better your question?
2d
comment How are asymptotes actually defined in rigorous mathematics?
@goblin Let me ask a question. Consider the set $\{(x,x \sin x) \mid x \in \mathbb{R}\}$. Would you like to give $y=x$ and $y=-x$ any role?
Aug
25
comment How are asymptotes actually defined in rigorous mathematics?
My notation intended to denote "real functions of a real variable", sorry for the confusion. In my experience $x \mapsto 1/x$ has exactly two asymptotes: $x=0$ and $y=0$. I have never seen a definition of asymptote for an arbitrary subset of $\mathbb{R}^2$. Your directed definition recalls of the definition of oblique asymptote, anyway.