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Results tagged with limits-and-convergence
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user 152618
Convergence of series, sequences and functions and different modes of convergence.
1
vote
0
answers
78
views
$ \varlimsup_{x\rightarrow 0^+}\frac{f(x)}{x^a}=\varliminf_{x \rightarrow 0^+}\frac{f(x)}{x^...
$f(x)$ is continuous for $\forall x \geq 0$ and monotonically decrease. $f(0)=0$. $a>0$. Is it true
$$ \varlimsup_{x\rightarrow 0^+}\frac{f(x)}{x^a}=\varliminf_{x \rightarrow 0^+}\frac{f(x)}{x^a} $$
…
- 419
1
vote
1
answer
53
views
Compare two limits related to Hölder condition
Suppose $f$ is a continuous function on $\mathbb{R}$. $0<a<1$. $B(x,r)$ is open ball centered at $x$ with radius $r$. Is it true that
$$ \varlimsup_{r\rightarrow 0} \frac{|f(x+r)-f(x)|}{|r|^\alpha} \l …
- 419
0
votes
1
answer
106
views
A problem of the limit of $\frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}...
Suppose that $f$ is a continuous function on $[0,1]$. For $0<a<1$, if
$$ \varlimsup_{\delta \rightarrow 0} \frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}} = \infty, $$
th …
- 419
6
votes
1
answer
459
views
A limit problem
Let $f$ be a bounded and continuous function, $0<a < 1$. $U(x,r)$ is the neighborhood of $x$ with diameter $r$. Can we prove the following equation of two limits
$$ \lim_{r\rightarrow 0} \sup_{y,z \i …
- 419