$g$ integrable over $[a,b]=> f(x)=int_{a}^{x}g$ is absolutely continuous on...
From the definition this flows naturally up until near the end: $forallepsilon>0$, I am looking for a $delta$ such that...
View ArticleExistance of a fixed point of a bijective, smooth function:
Let $[a,b],,, [c,d]$ be two bounded closed intervals of $mathbb{R}$ such that $$[a,b]cap[c,d]not=emptyset$$ and $f:[a,b]to[c,d]$ be a bijective, smooth function. We know that, if $[a,b]=[c,d],$ then...
View Articlecontinuous function and functional equation
Let $ g:mathbb{R} to mathbb{R}$ be a continuous function: $g(x) = g(x+1)$. Show that $g$ satisfies the equation: $g(x)=frac{1}{4} left(g(frac{x}{2})+g(frac{x+1}{2})right)$ for all $x$. Is it enough to...
View ArticleClosed subset of reals has continuous function vanishing on it [on hold]
Suppose $A subset mathbb{R}$ is closed. How can we show that there exists a continuous real-valued function $f$ such that $ker(f)=A$, i.e.e there is a function vanishing exactly on $A$?
View ArticleSuppose that $f$ is continuous and that $g circ f$ is differentiable. Must...
Suppose that $f$ is real function of a real variable defined on $(a,b)$ and that it is continuous on $(a,b)$. Suppose also that $g circ f$ is differentiable on the set $f((a,b))$ and that $g$ is not...
View ArticleHow do I find explicit formula of function, examine continuity and draw their...
Three-cylinder with height 4m and radii of the base 5, 3 and 1 m are going to put (in this order). Give an explicit formula for the following functions, you examine the functions on continuity and draw...
View ArticleProve that $phi : C[a,b]rightarrow mathbb{R}$ given by $phi(f)=int_a^bfdx$ is...
Prove that $phi : C[a,b]rightarrow mathbb{R}$ given by $phi(f)=int_a^bfdx$ is uniformly continuous. First, since $f$ is continuous $phi$ is well defined (integrals exist). Since $f$ is continuous on a...
View ArticleContinuously differentiable function defined through relation [on hold]
Suppose I have a function defined by $f(x,y)= a* f(f(x,x), f(y,y)) +b$ for $a>0$. I want to show that the sum of the partial derivatives at $(0,0)$ is nonnegative– how might I try to do this?
View ArticleContinuity and uniform continuity
I’m having trouble understanding the notion of uniform continuity, the definition states as follows: Let $f: Dtomathbb{R}$, $f$ is uniformly continous in $Xsubset D$ if: $$forallvarepsilon >0,...
View ArticleUse Cauchy-Schwartz inequality to prove that $ : mathscr H times mathscr H to...
Let $(a,b) in mathscr H times mathscr H$ be fixed. So we have to prove that for a given $epsilon gt 0$, we can find $delta_1 gt 0$ and $delta_2 gt 0$ such that $|<x,y>-<a,b>| lt epsilon$...
View ArticleContinuity of Popcorn Function (Thomae's Function)
I have to prove that the function $f:]0,1] rightarrow Bbb R$ : $$ 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 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 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 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 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 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 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 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 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 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 .
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