Looking 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 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 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:...
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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 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....
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 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 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 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 ArticleExplicit 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...
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