Explicit functions evaluated
(a) Defined $f$ by $f(y):=int_0^inftyfrac{xy}{(x^4+y^4)^{3/4}}dx$. Prove $f(y)$ is defined (i.e integral exists) for every $yinmathbb{R}$. (b)Prove that actually $f(y)=coperatorname{sign} y$ for some...
View ArticleExample of a function that converges to 0 pointwise but integral is 3/2?
Give an example of a sequence of continuous functions $(f_n)$, $f_n : [0, 1] to mathbb{R}$ that converges to zero pointwise, and such that the integral of each function within the given domain is...
View ArticleA smooth function which is nowhere real analytic, and preserves rationality...
There are examples $!^{[1]}$$!^{[2]}$ of continuous infinitely differentiable (class $C^infty$) functions $mathbb Rtomathbb R$ that are nowhere real analytic. I wonder if it is possible to construct...
View ArticleProve of continuity and open set
I need to prove this but I can’t figure out how. It would be nice if somebody can help me out with this . Let X and Y be nonempty subsets of $R^{N}$ and $R^{K}$, respectively. Prove the followings: A...
View Article$f$ is continuous, is $1/f$ continuous
Let $f: A rightarrow Bbb{R}$ be uniformly continuous. Suppose there exist $k>0$ s.t. $|f(x)| ge k$ for all $x in A$. Show that the function $1/f$ is also uniformly continuous on $A$. My proof Let...
View ArticleUsing definition of continuity to show that a function is continuous at the...
I am having some trouble with this particular type of question which asks, Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval....
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Random variables are determined by their characteristic functions proof
From the line $F_Y(t) = lim_{ntoinfty} F_y(t_n)$, here the author is assuming that $t$ is not a continuity point of $F_Y$, and is being approximated by continuity points $t_n$, and the result holds as...
View Articleif $|f(k) | le k$ for all integers $k$, does that mean $ |f(x)| le |x|$ for...
if $|f(k) | le k$ for all integers $k$, does that mean $ |f(x)| le |x|$ for all $x$ in $mathbb R$? Note that $f$ is uniformly continuous. This question is a follow up to a previous answer:...
View ArticleVerifying if a function is a.e. equal to a continuous function then it is...
Is it true that if a function is almost everywhere equal to a continuous function then it is continuous almost everywhere? I know that the converse is false. i.e. if f is continuous a.e. , there must...
View ArticleLooking for an example of an infinite metric space $X$ such that there exist...
I am looking for an example of an infinite metric space $X$ such that there exist a continuous bijection $f: X to X$ which is not a homeomorphism . Please help . Thanks in advance .
View ArticleRiemann type function continuity
Define real-valued function $f$ on $mathbb{R}cap[0,1]$ by setting $$f(x)= begin{cases} x,,,text{if $x$ irrational}\ psin(frac{1}{q}),,, text{if $x=frac{p}{q},gcd(p,q)=1$}\ end{cases} $$ Prove $f$ is...
View ArticleIf $f$ is continuous on $[0,∞)$, $f(0) = 0$, and $lim_{x→∞} f(x) = L > 0$,...
I am taking a practice exam for an analysis exam, and am stuck on how to do this problem.
View ArticleContinuity of a function and its derivative
Does $f’$ being continuous on $(a,b)$ imply that $f$ is also continuous on $(a,b)$? I think this is probably quite a straightforward question but it’s key to solving the problem that I’m working on.
View ArticleContinuity of $1/f(x)$
Suppose that $f(x)$ is continuous on $[a, b]$ and that $f(x) geq c > 0$ for some constant $c$. Prove that $1/f(x)$ is continuous on $[a, b]$.
View ArticleHow to show map is non-singular
Let $f:;mathbb{R}^ntomathbb{R}^n$ be differentiable. Suppose that for all $xinmathbb{R}^n:$ $$lVert mathrm{D}f(x)-mathrm{I}rVertleq frac{1}{2}$$ where $lVertcdotrVert$ is the operator norm. I need to...
View ArticleShow that $Y_1[t]- Y_2[t] to 0$ as long as $t to infty$ – Differential equations
Let the differential equation $L[y] = a y” + by’ + cy = g(t)$, where $a$, $b$ and $c$ are strictly positive numbers. If $Y_1(t)$ and $Y_2(t)$ are solutions at the $L[y]$ equation, show that $Y_1[t]-...
View ArticleHow can I show that this function is discontinuous at the point $x=1$?
Suppose you had the function $$ f(x) = ; text{ the integer part of } x $$ I wish to show that this is not continuous at the point $x=1$, which I will try to do by showing that $lim_{x rightarrow 1}...
View ArticleA function $f(x)$ is continuous in the interval $[0,2]$. It is known that...
(A) There exists a $y$ in the interval $(0,1)$ such that $f(y)=f(y+1)$. (B) For every $y$ in the interval $(0,1), f(y) = f(2−y)$. (C) The maximum value of the function in the interval $(0,2)$ is $1$....
View ArticleContinuity of Popcorn Function (Thomae's Function)
I have to prove that the function $f:]0,1] rightarrow Bbb R$ : $$ f(x) = begin{cases} frac1q, & text{if $x in Bbb Q$ with $ x=frac{p}q$ for $p,q in Bbb N$ coprime} \ 0, & text{if $x notin Bbb Q...
View ArticleUse Cauchy-Schwartz inequality to prove that $ : mathscr H times mathscr H to...
Let $(a,b) in mathscr H times mathscr H$ be fixed. So we have to prove that for a given $epsilon gt 0$, we can find $delta_1 gt 0$ and $delta_2 gt 0$ such that $|<x,y>-<a,b>| lt epsilon$...
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