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"Nothing is more important than to see the sources of invention which, in my opinion, are more interesting than the inventions themselves."

--Gottfried von Leibniz (1646-1716)


58m
answered Questions about $L^p$ spaces and convergences
2h
comment Infinite prime numbers
See the first answer here.
18h
comment Would a function $f: [0,1]\to\mathbb{R}$ that satisfies Intermediate Value Theorem be continuous?
And also this to answer your question.
18h
comment Would a function $f: [0,1]\to\mathbb{R}$ that satisfies Intermediate Value Theorem be continuous?
See this, though.
18h
comment If a function is bounded and the variable is bounded, is the function continuous?
No.${}{}{}{}{}$
21h
comment Finding a closed subset in $\mathbb{R}^2$ such that its image is not closed in $\mathbb{R}$
The graph of $y=1/x$.
21h
comment Cardinality of infinite between the set of rationals and set of reals
Google "Continuum hypothesis".
23h
comment Given $f_n \rightarrow f$ a.e. Does $||f_n||_p \rightarrow ||f||_p$ imply $f_n\rightarrow f$ in $L^p$?
Oops, I didn't see you were asking for a proof verification.
1d
comment Common logarithm question
Can you write $10/\root3\of {10}$ as $10^x$ for some $x$? If so, what would $\log 10^x$ be?
1d
comment If a continuous function $f$ on $[a, b]$ is differentiable and $f'\in L^1[a, b]$, can we conclude that $f$ is absolutely continuous?
Yes. For a reference, see my answer here.
1d
answered Does a nondecreasing, differentiable function have continuous derivative?
2d
comment Derivative changes sign for continuous and differentiable function
You should (in my opinion) also give the proof for the case $f(b)<f(a)$, since then it is seen the hypothesis that $f'(b)>0$ is needed.
2d
comment Derivative changes sign for continuous and differentiable function
@G.T.R If $h(a+\epsilon)\ge h(a)$ for all $\epsilon>0$, then ${h(a+\epsilon)-h(a)\over \epsilon}\ge0$ for all $\epsilon>0$. It then follows that $h'(a)$ can't be negative.
2d
comment Derivative changes sign for continuous and differentiable function
This looks fine. Your statement is a particular case of Darboux's Theorem.
2d
comment Does a nondecreasing, differentiable function have continuous derivative?
I presume the OP wants $f$ to have a finite derivative for all $x\in[0,1]$.
2d
revised Getting a diverse set of three numbers from two numbers
edited tags
2d
comment Example of a continuous function that is monotone at no point
It seems Andres Caicedo's answer there contains what you're looking for.
Jul
26
awarded  Nice Answer
Jul
25
revised Proof that there is a bijection, if there are injective maps in both directions
deleted 29 characters in body
Jul
25
answered Proof that there is a bijection, if there are injective maps in both directions