What is Continuous functions: Definition and 135 Discussions
In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.
Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.
As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M(t) denoting the amount of money in a bank account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.
Hello.
How to prove that in smooth infinitesimal analysis every function on R is continuous? (Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.)
Thanks.
Let ##f : \mathbb{R} \to \mathbb{R}## be a uniformly continuous function. Show that, for some positive constants ##A## and ##B##, we have ##|f(x)| \le A + B|x|## for all ##x\in \mathbb{R}##.
Suppose f is a function such that f'(7) is undefined. Which of the following statements is always true? (Give evidences that supports your answer, then explain how those evidences supports your answer)
a. f must be continuous at x = 7.
b. f is definitely not continuous at x = 7.
c. There is not...
##f## is continuou on ##\mathbb{C}##, so for al ##\epsilon>0##, there is a ##\delta>0## such that $$|\tilde{z}-z|\leq \delta \Rightarrow |f(\tilde{z})-f(z)|\leq \epsilon$$ for all ##\tilde{z}## and ##z## in ##\mathbb{C}##.
Complex conjugation is a norm preserving operation on ##\mathbb{C}##, so...
Suppose ##f## is not uniformly-continuous. Then there is ##\epsilon>0## such that for any ##\delta>0##, there is ##x,y\in K## such that if ##|x-y|<\delta##, ##|f(x)-f(y)|\geq \epsilon##.
Choose ##\delta=1##. Then there is a pair of real numbers which we will denote as ##x_1,y_1## such that if...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...
I need some help in understanding the proof of Proposition 3.12...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...
I need some help in understanding the proof of Proposition 3.12...
Definition: A function f mapping from the topological space X to the topological space Y is continuous if the inverse image of every open set in Y is an open set in X.
The book I'm reading (Charles Nash: Topology and Geometry for Physicists) emphasizes that inversing this definition would not...
In analysis, the pasting or gluing lemma, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. Can we have a similar situation for uniform continuous functions?
Homework Statement
The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong.
Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...
My question is maybe elementary but I don't know the answer. I have a function f absolutely continuous in (a,c) and in (c,b), f continuous in c. Is f absolutely continuous in (a,b)?
I think the answer is negative but I can't find a counterexample. I really apreciatte your help.
I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...
I am focused on Chapter 1: Continuity ... ...
I need help with an aspect of Example 1.3.8 ...
Duistermaat and Kolk"s Example 1.3.8 reads as follows:In the above example we read the...
I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...
I am focused on Chapter 5: Continuous Functions ...
I need help in fully understanding an aspect of the proof of Theorem 5.3.2 ...Theorem 5.3.2 and its proof ... ... reads as follows:In...
I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ...
I am focused on Chapter 5: Continuous Functions ...
I need help in fully understanding an aspect of Example 5.1.6 (h) ...Example 5.1.6 (h) ... ... reads as follows:
In the above text from...
I've learned that composition of continuous functions is continuous. ##\log x## and ##|x|## are continuous functions, but it seems that ##\log |x|## is not continuous. Is this the case?
Homework Statement
Let ##C## be the space of continuous real functions on ##[0,\pi]##. With any function ##f\in C##, associate another function ##g=T(f)## defined by $$g=T(f)\equiv \int_0^\pi \cos(t-\tau) f(\tau) \, d \tau$$
a) Show ##T## is a linear transformation from ##C## to ##C##.
b)What...
I am working my way through elementary topology, and I have thought up a theorem that I am having trouble proving so any help would be greatly appreciated.
----------------------
Theorem: Let A ⊂ ℝn and B ⊂ ℝm and let f: A → B be continuous and surjective. If A is bounded then B is bounded...
Homework Statement
The problem is posted below in the picture. I looked at c and d and can do those. I am unsure about a and b.
Homework EquationsThe Attempt at a Solution
I looked at graphing the problems, but I think it is a wrong approach.
I have been working on this exercise 5 and kind of stuck how to start the problems. I would think to start with a graph, but I feel this is wrong. I am just stuck on a and b.
Homework Statement
Prove $$T\int_c^d f(x,y)dy = \int_{c}^dTf(x,y)dy$$ where $$T:\mathcal{C}[a,b] \to \mathcal{C}[a,b]$$ is linear and continuous in L^1 norm on the set of continuous functions on [a,b] and
$$f:[a,b]\times [c,d]$$ is continuous.
Homework EquationsThe Attempt at a Solution
[/B]...
Consider two functions f, g that take on values at t=0, t=1, t=2.
Then the total error between them is:
total error = mod(f(0)-g(0)) + mod(f(1)-g(1)) + mod(f(2)-g(2))
where mod is short for module.
This seems reasonable enough.
Now, consider the two functions to be continuous on [0,2].
What...
I noticed that uniform continuity is defined regardless of the choice of the value of independent variable, reflecting a function's property on an interval. However, if on a continuous interval, the function is continuous on every point. It seems that the function on that interval must be...
The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom.
Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map.
What I understand:
Let ##X## be a finite product space and ##...
I claim that if a function ##f:\mathbb{R}\rightarrow\mathbb{R}## is continuous at a point ##a##, then there exists a ##\delta>0## and ##|h|<\frac{\delta}{2}## such that ##f## is also continuous in the ##h##-neighbourhood of ##a##.
Please advice if my proof as follows is correct.
Continuity at...
Homework Statement
A function f is defined as follows:
ƒ(x) = sin(x) if x≤c
ƒ(x) = ax+b if x>c
Where a, b, c are constants. If b and c are given, find all values of a (if any exist) for which ƒ is continuous at the point x=c.
Homework EquationsThe Attempt at a Solution
I was unsure of how to...
Hi, say X is a topological space with subspaces Y,Z , so that
Y and Z are homotopic in X. Does it follow that there is a continuous
map f:X→X with f(Y)=Z ? Do we need isotopy to guarantee the existence of
a _homeomorphism_ h: X→X , taking Y to Z ?
It seems like the chain of maps...
Actually, the theorem is that functions that are uniformly continuous are Riemann integrable, but not enough room in the title!
I'm failing to see the motivation behind proof given in my lecturer's notes (page 35, Theorem 3.29) and also do not understand the steps.
1) First thing I'm...
Homework Statement
Suppose that k(t) is a continuous function with positive values. Show that for any t (or at least for any t not too large), there is a unique τ so that τ =∫ (k(η)dη,0,t); conversely
any such τ corresponds to a unique t. Provide a brief explanation on why there is such a...
Homework Statement
How many continuous functions f are there which satisfy the equation (f(x)^2) = x^2 for all x? Homework Equations
The Attempt at a Solution
What method should I use to solve this? Is there a specific strategy involved besides plug and chug? Off the top of my head, I can only...
Hi. Could help me with the following problem?
Let f be a real function, increasing on [0,1].
Does there exists a sequence of functions, continuous on [0,1], convergent pointwise to f? If so, how to prove it?
I would really appreciate any help.
Thank you.
Let U be a subset of ℝn be an open subset and let f:U→ℝk be a continuous function.
the graph of f is the subset ℝn × ℝk defined by
G(f) = {(x,y) in ℝn × ℝk : x in U and y=f(x)}
with the subspace topology
so I'm really just trying to understand that last part of this definition...
Working on some problems that have vectors, for example
f(x) = [-x1/|X|3, -x2/|X|3]
And then I am asked to find the largest interval of existence. The answer says "E = R2 ~ {0}.
I'm not sure what this means. Does it mean the interval of existence is everywhere except 0? Is that what the ~...
If every continuous function on M is bounded, what does this mean?
I am not sure what this function actually is... is it a mapping from M -> M or some other mapping? Is the image of the function in M? Any help would be greatly appreciated!
Homework Statement
A function f:[a,b] \rightarrow ℝ is called piecewise continuous if there exists a finite number of points a = x0 < x1 < x2 < ... < xk-1 < xk = b such that
(a) f is continuous on (xi-1, xi) for i = 0, 1, 2, ..., k
(b) the one sided limits exist as finite numbers
Let V be the...
Let f and g be two continuous functions on ℝ with the usual metric and let S\subsetℝ be countable. Show that if f(x)=g(x) for all x in Sc (the complement of S), then f(x)=g(x) for all x in ℝ.
I'm having trouble understanding how to approach this problem, can anyone give me a hint leading me...
My problem is as follows (sorry, but the tags were giving me issues. I tried to make it as readable as possible):
Let X have the pdf f(x)= θ * e-θx, 0 < x < ∞
Find pdf of Y = ex
I've gone about this the way I normally do for these problems.
I have
G(y) = P(X < ln y) = ∫ θ * e-θx...
Say I have a function F(x,y)=(f(x),g(y)), F:X×Y→X'×Y'. Is there a theorem that says if f:X→X' and g:Y→Y' are continuous then F(x,y) is continuous. I've proved it, or at least I think I have, but I'd like to know for sure whether or not I'm right.
I know that its not necessarily true that a...
Homework Statement
Let X and Y be metric spaces such that X is complete. Show that if {fα(x) : α ∈ A} is a bounded subset of Y for each x ∈ X, then there exists a nonempty open subset U of X such that {fα(x) : α ∈ A, x ∈ U} is a bounded subset of Y.
Homework Equations
Definition of...
Let X be a uniform integrable function, and g be a continuous function. Is is true that g(X) is UI?
I don't think g(X) is UI, but I have trouble finding counter examples.
Thanks.
Homework Statement
Let ##A \subset X##; let ##f:A \mapsto Y## be continuous; let ##Y## be Hausdorff. Show that if ##f## may be extended to a continuous function ##g: \overline{A} \mapsto Y##, then ##g## is uniquely determined by ##f##.
Homework Equations
The Attempt at a Solution...
Homework Statement
Continuous function f: R → R, f(x) = 1 - e(x)sin(x)
Continuous function g: R → R, g(x) = 1 + e(x)cos(x)
Homework Equations
Using Rolle's Theorem, prove that between any two roots of f, there exists at least one root of g.
The Attempt at a Solution
I think I'm meant...
In analogy to vector spaces, can we define a set of "basis functions" from which any continuous function can be written as a (possibly infinite) linear combination of the basis functions?
I know the trigonometric functions 1, sin(nx), cos(nx) can be used for monotonic continuous functions...
Hey guys,
I have been working on the following question:
http://imageshack.us/a/img407/4890/81345604.jpg
For part a
f and g are continuous on I
=> there exists e > 0 and t_0 s.t.
0<|{f(t) - g(t)} - {f(t_0) - g(t_0)}| < e
using |a-b| >= |a| - |b|,
|{f(t) - g(t)} -...
Homework Statement
Determine all continuous functions g: R -> R such that g(x + y) = g(x) + g(y) for all x, y \in \mathbf{R}
The Attempt at a Solution
g(x) = g(x + 0) = g(x) + g(0). Hence G(0) = 0.
G(0) = g(x + -x) = g(x) + g(-x) = 0. Therefore g(x) = -g(-x).
It seems obvious that the only...
Homework Statement
Take f: (a,b) --> R , continuous for all x0in (a,b)
and take (Ω = (a,b) , F = ( (a,b) \bigcap B(R)) where B(R) is the borel sigma algebra
Then prove f is a borel function
The Attempt at a Solution
I know that continuity of f means that for all x in (a,b) and all...
Homework Statement
If C^1(\mathbb T) denotes the space of continuously differentiable functions on the circle and f \in C^1(\mathbb T) show that
\sum_{n\in\mathbb Z} n^2 |\hat f(n)|^2 < \infty
where \hat f(n) is the Fourier coefficient of f.
The Attempt at a Solution
Since f is...
Hi, can someone give me pointers on this question
Homework Statement
Prove or provide a counterexample: If f : E -> Y is continuous on a
dense subset E of a metric space X, then there is a continuous function
g: X -> Y such that g(z) = f(z) for all z element of E.
The Attempt at a Solution...