Fundamental Theorem of Calculus application for $f(x)geq 0$
Can anybody help me with how to solve the following question using the fundamental theorem of calculus? I’m a bit confused… Suppose that $f$ is a continuous function on $[a, b]$ and that $f(x)geq 0$...
View ArticleEvery uniformly continuous real function has at most linear growth at infinity
Assuming $f:mathbb Rtomathbb R $ be an uniform continuous function, how to prove $$exists a,bin mathbb R^+ quad text{such that}quad |f(x)|le a|x|+b.$$
View ArticleContinuity and Uniform Continuity on half closed intervals
I have been stuck on the following problem for a long time : Prove that if a function $f:(a,b]tomathbb R$ is continuous, then it is uniformly continuous if and only if $lim_{xto a^+}f(x)$ exists and is...
View ArticleConceptual question regarding convergence and continuity
Let $X,Y$ be metric spaces. Let $(f_n)$ be a sequence of functions from $X$ to $Y$ equicontinuous that converges pointwise to a function $f:X to Y$. Then, ${f,f_1,f_2,…}$ is equicontinuous. When...
View Articlefunction of 2 variables
we have the next function: $$f(x,y)=begin{cases} dfrac{sqrt{x^2y^2+1}-1-x^2-y^2}{x^2+y^2} & (x,y)neq (0,0) \ c & (x,y)=(0,0) end{cases}$$ Is there $c$ that $f(x, y)$ is continuous function...
View ArticleEvery continuous map of a closed interval into itself has a fixed point
The Question: Please show this theorem: Let $f: I=[a,b] rightarrow mathbb{R}$ be a continuous map such that $f(I) supset I $. Then $f$ has a fixed point on I. My Attempt: Suppose there is a function,...
View ArticleStrictly monotonic increasing function with a closed domain and range
Let $a,b,c,d in mathbb{R}$ with $a<b$, $I = [a,b]$. Let $f: I rightarrow mathbb{R}$ be a monotonic, strictly increasing function. Also $c<d$ and $f([a,b]) =[c,d]$ a) Proof that $f$ is continuous...
View ArticleThe image of the inverse of a continuous function
First of all I’m not sure if my title is correct with the question, I find it hard to really get about what kind of set this question is about. It would be very helpful if someone could explain this to...
View Article$f$ is monotone on D and $f(D)$ is an interval
$f$ is monotone on D and $f(D)$ is an interval then $f$ is continuous Is my proof right? pf) First, suppose it is monotone increasing Since $f(D)$ is an interval there is $[c,d]$ such that $f(x)in...
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Help with proof; Pre-image of a continuous function around a maximum is open.
Suppose a function $u$ is defined in an open and connected set $D$ and has maximum value $c$. Then if $u$ is not constant in D then the set ${u(z) < c mid z in D}$ is non-empty and open. This was...
View ArticleReverse Intermediate Value Theorem
What does it mean to say that a real valued function $ f : [a, b] rightarrow mathbb{R} $ is continuous at $ x_0 in [a, b] $? Assume that $ f : [a, b] rightarrow mathbb{R} $ is continuous State, without...
View ArticleUsing $epsilon$-$delta$ argument to show continuity
Show using the $epsilon$-$delta$ definition of continuity that $f(x)=begin{cases} 11&text{if}~0leq xleq 1\x&text{if}~ 1<xleq 2end{cases}$ is continuous on $[0,1)cup (1,2]$ How do we...
View Articlecontinuous (smooth) maps and group homomorphism
Consider a topological group $G$ (or smooth Lie group) and a topological space $M$ (or smooth manifold) and a group homomorphism $phi:Grightarrow Sym(M)$, where $Sym(M)$ is the symmetry group of M,...
View Article$f in C[a,b]$ be such that $int_c^d f(x)dx=0 , forall c,d in [a,b] , c
Let $f:[a,b] to mathbb R$ be a continuous function such that $int_c^d f(x)dx=0 , forall c,d in [a,b] , c<d$ ; then is it true that $f(x)=0 , forall x in [a,b]$ ?
View ArticleProve that $g$ is continuous at $x=0$
Given, $g(x) = frac{1}{1-x} + 1$. I want to prove that $g$ is continuous at $x=0$. I specifically want to do an $epsilon-delta$ proof. Related to this Is $g(x)equiv f(x,1) = frac{1}{1-x}+1$ increasing...
View ArticleWhich of these conditions on continuity imply that two topologies are equal?
Suppose $X$ is a topological space and $mathcal{T}$, $mathcal{T}’$ are two topologies on $X$. Determine whether or not the given condition implies $mathcal{T} = mathcal{T}’$. If not, give a...
View Articlesuppose $x_n = frac{p_n}{q_n} in Q_n$ where $frac{p_n}{q_n}$ is in reduced...
suppose $x_n = frac{p_n}{q_n} in Q_n$ where $frac{p_n}{q_n}$ is in reduced form and $x_n to a, a notin mathbb{Q}$. Prove $q_n to infty$ and use the result to deduce the Thomae function is continuous at...
View ArticleShow the map from $0-1$ sequences to the corresponding binary numbers is...
Show that the function: $$f : { 0,1}^{mathbb N} to [0,1], f(a_1, a_2, cdots, ) = 0.a_1a_2cdots $$ is continuous. There’s hint from: Prove a map of binary expansion is continuous but don’t know what to...
View ArticleFor a > 0, consider any continuous function f : [−a, a] → R such that f(−a) =...
Im having a bit of trouble getting an answer for this question: For a > 0, consider any continuous function f : [−a, a] → R such that f(−a) = f(a). Show that there exists x ∈ [−a, 0] such that f(x)...
View ArticleUnderstanding the notation $fcolon Dsubseteqmathbb{R}tomathbb{R}$ in the...
In brief, I’ve had virtually no mathematical education prior to seven months ago. What I did know had largely been the result of watching Khan Academy. Seven months ago I began a precalculus textbook,...
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