Answered: Let f be a function defined from a…

Let f be a function defined from a topological space (X, T) to atopological space (Y,T,).Let a be a point in X such that {a} is open in T,. Show that f is continuous at a

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...

See similar textbooks

Related questions

Q: Define an inner product space for continuous functions on [0, 1] as (S, g) = [f(x)g(x)ax with the…

Q: Let f:R→R satisfy f(x+y) =f(x) +f(y), ∀x,y∈R. If limx→0f(x) exists, prove that f is continuous on R

Q: Let f be a bounded function on [a,b] for which "f = ["f. Prove that f E R[a,b]. (Hint: Use the…

Q: Let A = {0, 1} with the discrete topology and let f: A R be defined by f (0) = -1, f(1) = 1.

Q: Let IR have its usual topology and let Y = {x, y, z}. Define f: R → Y by: x 0 (x, f(x) y, z, Is Y…

A: Quotient topology: If X is topological space and A is a setand if p:X→A is surjective map, then…

Q: is continuous at (0,0). 3. Show that X = [0, 1] with the discrete metric is bounded but not totally…

Q: Let f function from (R,d ) to (R,d1) defined as d(x) = x , where d discrete metric and d1 .is the…

Q: Theorem : Let f be defined on [a, b] , if f has a local minimum at a point x € (a, b), and if f'(x)…

Q: Let f be continuous on [a, b] and differentiable on (a, b). If there exists c in (a, b) such that…

A: The converse of the given statement is true but not the given statement. Counter Example: Consider…

Q: f If Xo's Compact, x=y Continuous bijective L Topelagied space Then f is homeomorphism

Q: Let F be a nonempty set of functions that map [0,1] into [0,1]. For f and g in F, define…

A: Given F be a nonempty set of the function that map 0,1 into 0,1. And for f and g in F define…

Q: Let A = {0, 1} with the discrete topology and let f : A → R be defined by f(0) = −1, f(1) = 1. Show…

Q: Prove the following theorem: If ƒ(x, y) is defined in an open region R of the xy-plane and if ƒx and…

Q: Let f: U C be a holomorphic function, where UCC be an open subset in the upper half-plane. Let V =…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let a < b be real numbers, and let f be a continuous function defined on I = a, b. Then f is Riemann…

Q: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = %3D (x, y) = (0,0) x²y+2xy² Let…

Q: Let X be a topolegical space be a and let a Continuous map. Define the Set Y=3x€ X : flx) = x} X is…

Q: Let f be a continuous function with domain R. Compute the limit (z- t) f(t) lim dt.

Q: Let f : X → Y be a bijective continuous function. If X is compact and Y is Hausdorff then f is a…

A: To prove , Let f: X→ Y be a bijective continuous function. If X is compact and Y is Hausdorff then f…

Q: Let f: X_nte, Y be X,Y are Tj spaces. show that: Xö locally compact iff Y is so . perfect function,…

A: Given: f:X→ontoY is a perfect function, where X,Y are T2-spaces. To show that X is locally compact…

Q: Let f be a bounded function on [a, b]. If P and Q are partitions of [a, b], prove that L( f , P) ≤…

A: Given f be a bounded function on a,b. If P and Q are the partitions of a,b. Then We have to prove…

Q: Let f:X Y be function and g:X→fcXXY, g(x) = (x, f(x)). Prove that if g is a homeomorphism then f is…

A: Given that, f:X→Y be function and g:X→f⊂X x Y defined byg(x) = (x,f(x))Given that, 'g' is…

Q: Let X be the set of all continuous functions defined on the interval [a,b]. Let d: X × X →R be a…

A: We need to prove the all three conditions for metric.

Q: Let f be a continuous mapping of a metric space X into metric space Y,g be a continuous mapping of…

Q: Let X ų be topological spaces and f: X→Y be a continuous map. If X is completely normal then Y is…

Q: Let f be a function defined on R satisfying If(z) - f(y)| < 히 - J), V,yER. a. Show that f is a…

A: Given that f be a function defined on ℝ satisfying…

Q: ht x and y be normed spaces. Show that a Lincar operator T:Xy is bounded If and only T maps bounded…

Q: Let f be a bounded function on [a,b], and suppose that (PnnEN is a family of partitions on [a,b]…

A: Given that f is a bounded function on [a,b]. (Pn)n∈ℕ is a family of partitions on [a,b] Assume that…

Q: Let F be a nonempty set of functions that map [0, 1] into [0, 1] . For all f and g in F, define…

A: Consider the set F={f: f:[0, 1]→[0, 1] } d is defined as d(f, g)=Sup{|f(x)-g(x)|} To Show : d is a…

Q: Let Y be any topological space and y: [0, 1] → Y be a continuous function so that r(0) = r(1) = xp.…

Q: Let f be a bounded function on [a,b] and suppose that (P)aEN is a family of partitions on [a,b]such…

A: Given: Let f be a bounded function on [a,b] and (Pn)n∈N is a family of partitions on [a,b] such that…

Q: Let f be a mapping from [1,+[ to [1,+[, defined by f(x)=x+1/x. Then * O None of the choices O fis…

Q: Prove that f(2) = z/(z* + 1) is continuous at all points inside and on the unit circle Iz] = 1…

A: we need to find those points where the denominator ( z^4+1) equals to 0

Q: Construct a one-to-on continuous mapping of the set {(#1, 12, - ..., In); 0 < T; < 1. j= 1, 2,…

A: Given that : x1,x2,x3,.........,xn, 0<xj<1, j=1,2,3,.....,n So we have to take ,…

Q: ) Let f be a continuous function that maps a finite closed interval [a, b] into itself. Prove that f…

Q: Every continuous function from normed space X into normed space Y is bounded linear operator True…

Q: Let f: R → R with the usual topology that sends compact sets to compact sets in R ^ 2 Is the…

A: For any x,y∈R if y=fx there there exists a number x±∈R in neighborhood of x such that y±=fx±.

Q: Suppose that ƒ is a Riemann integrable function on [a, b]. Prove, in two lines only, that if f is…

A: Bounded function

Q: Let (X, d₁) and (Y, d₁) be metric spaces. Let f: X Y be continuous function, then f-¹(G) is open in…

A: Solution

Q: a function f is continuous at x0 element of X if and only if f is both lower semi continuous and…

Q: Find and describe the points in the space that the function f(x, y,z)=In(xyz) is continuous.

A: We have to find and describe the points in the space that the function f(x,y,z)=ln(xyz) is…

Q: Let f : AC X → Y be a continuous function between metric spaces. If A is open then so is f(A).

A: Continuous Function: Let (X, dX) and (Y, dY ) be metric spaces. A function f : X → Y is continuous…

Q: Exercise 6. Prove that if f [a, b] : interval. → R is continuous, then f([a, b]) is a closed and…

A: 6. Given that f : a,b→ℝ is a continuous function. To find: f a,b is closed and bounded interval.…

Q: 2. Let V be the vector space of all continuous functions defined on the interval Determine whether…

Q: Let X and Y be two topological sPaces. Jet A and B CX be such that Ā =B. et f:X y be a continuous…

A: Given: X and Y be two topological spaces Let A and B ⊂X be such that A=B

Q: Let f : [a, b] → R be a strictly decreasing function. Prove that f is integrable [a, b). on

Q: Show that the space C(R) of all continuous functions defined on the real line is an…

A: According to the given information, it is required to show that C(R) of all continuous functions…

Q: Assume that f is continuous on [a, b] and differentiable on (a, b). Prove that if f' > 0 for all a <…

A: Assume that f is continuous on [a,b] and differentiable on (a,b). Given that f'>0 for all…

Question

100%

Let f be a function defined from a topological space (X,
T) to atopological space (Y,T,).Let a be a point in X such
that {a} is open in T,. Show that f is continuous at a

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Complexity$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
Let f:X->Y be a function between metric spaces (X,d) and (Y,d). Prove that f: (0, infinity) -> R, f(x) is not uniformly continuous.
Let f:X->Y be a function between metric spaces (X,d) and (Y,d). Prove that f: (0, infinity) -> R, f(x)=1/x is not uniformly continuous.
Suppose that (X,d) is a metric space and R has the standard metric. Let f : X → R be a continuous function. For each r > 0, show that A={x∈X| |f(x)|<r} is open in X.
Let F be a nonempty set of functions that map [0,1] into [0,1]. For f and g in F, define d(f,g)=sup{|f(x)-g(x)|:x in [0,1]} Show that d is a metric on f.
Let f be a continuous mapping of a metric space X into metric space Y,g be a continuous mapping of metric space Y into metric space Z. Then gof ia a continues mapping of X into Z
Let f : [0,∞) → R. Assume that f is uniformly continuous on [0, 1] andon [1,∞). Show that it is uniformly continuous on [0,∞)
is g continuous on (0,1]?
Give an example if f : (0,1) -> R that is bounded, continuous but not uniformly continuous on (0,1).
Let f: R->R be a continuous function that is convex on (-∞), 0] and [0, ∞)) and has a local maximum at the point 0. Prove that the function f is not differentiable at the point 0.
Show that if f is Riemann integrable on [a,b] and f(x) ≥ 0 for all x ∈ [a,b],then
Let (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If (Y, T1) is a T1 space, then (X, T) is a T1 space?