# What is Continuous: Definition and 1000 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account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.

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1. ### Find f s. t. ||f||=1 and f(x) < 1 with ||x||=1

Let ##f## be a continuous function defined in ##\mathbb{R}^n##. ##||\cdot ||## is the standard Euclidean metric. Then here are my suggested ways to choose ##f##: 1. Choose any continuous ##f## that satisfies $$1=\sup_{||x||\leq 1}||f||\neq \max_{||x||\leq 1}||f||$$ because the inequality...
2. ### Continuous functions on metric spaces part 2

Hi, The task is as follows For the proof I wanted to use the boundedness, in the script of my professor the following is given, since both ##(X,d)## and ##\mathbb{R}## are normalized vector spaces I have now proceeded as follows ##|d(x,p)| \le C |x|## according to Archimedes' principle, a...
3. ### B Finite linear combination of continuous functions is continuous?

##G## and ##H## are real valued Lipschitz continuous functions. There exists a ##K_1,K_2\geq 0## such that for all ##s,t##, $$(s-t)^2\leq K_1^2 (G(s)-G(t))^2$$ and $$(s-t)^2\leq K_2^2 (H(s)-H(t))^2.$$ Is ##aG(t)+bH(t)## where ##a,b## are real constants also Lipschitz continuous? I tried showing...
4. ### Wavelet transform (CWT and DWT)

Hello, I recently got interested in wavelets. The main idea seems clear: we compute the inner product between the signal ##x(t)## and a chosen wavelet for different scale factors and translations of the wavelet over the signal. The inner product provides the coefficient for a wavelet with a...
5. ### I Expressing any given point on plane with one unique number

Currently, as far as I know, the two main ways to express any given point on a plane is through either cartesian plane or polar coordinates. Both of which requires an ordered pair of two numbers to express a point. However, I wonder if there exists such a system that could express any given...
6. ### POTW A Function in the Continuous Hölder Class

Let ##0 < \alpha < 1##. Find a necessary and sufficient condition for the function ##f : [0,1] \to \mathbb{R}##, ##f(x) = \sqrt{x}##, to belong to the class ##C^{0,\alpha}([0,1])##.
7. ### I Topological argument principle

φ is a continuous, proper and locally one-to-one map.Is it a globally one-to-one map?
8. ### POTW Hölder Continuous Maps from ##R## to a Metric Space

Let ##\gamma > 1##. If ##(X,d)## is a metric space and ##f : \mathbb{R} \to X## satisfies ##d(f(x),f(y)) \le |x - y|^\gamma## for all ##x,y\in \mathbb{R}##, show that ##f## must be constant.
9. ### Showing piece-wise function continuous

For this, , The solution is, However, should they not write ##f(x) = \cos x## on ##[\frac{pi}{4}, \infty)## Many thanks!
10. ### I Is there a Boltzmann distribution for a system with continuous energy?

Hi. I'm not sure where to put this question, thermodynamics or the quantum physics forum (or somewhere else). For a system in equillibrium with a heat bath at temperature T, the Boltzman distribution can be used. We have the probability of finding the system in state n is given by ##p_n =...
11. ### Proving g(x) is continuous over interval (-∞,-2)

For number 18, The solution is, However, should they not write "For ## -∞ < a < -2##" since ##a ≠ -∞## (infinity is not a number)? Many thanks!
12. ### Confused about Continuous Endpoints: -1 < a < 1?

For this problem, I don't understand why they are saying ##-1 < a < 1## since they are trying to find where ##f(x)## is continuous including the endpoints ##f(-1)## and ##f(1)## Why is it not: ##-1 ≤ a ≤1## Many thanks!
13. ### I Proving Continuous Functions in Smooth Infinitesimal Analysis

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.
14. ### I Can a Satellite Maintain its Angular Velocity with Continuous Low Thrust?

Suppose two satellites are in a circular heliocentric orbit with radius R and with angular velocity O'. Satellite 2 then undergoes a low continuous thrust. Can Satellite 2 (the one that undergoes the continuous low thrust) maintain the same angular velocity O' about the sun? It seems that...
15. ### POTW Uniformly Continuous Functions on the Real Line

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}##.
16. ### Continuous Optimization, is this convex?

f(x)=ln(|x1|+1)+(-2x1 2 +3x2 2 + 2x3 3) + sin(x1 + x2 + x3), for this problem in particular would be it be sufficient to find the Hessian and to see if that matrix is semi positive definite to determine if it convex?

29. ### MHB Interpolating Points with Continuous Modular Functions?

Define a continuous function F(x;n) that interpolates points (x, x mod n) for a given integer n and all integer x. For example F(x;2)=\frac{1}{2}-\frac{1}{2}\cos\left(\pi x\right) interpolates all points (x, x mod 2) when x is an integer. Similarly F(x;3) should interpolate points (0,0), (1,1)...
30. ### Prove that the concatenation function is continuous

Let f be continuous in [0,1] and g be continuous in [1,2] and f(1)=g(1). prove that $$(f*g)= \begin{cases} f(t), 0\leq t\leq 1\\ g(t), 1\leq t \leq2 \end{cases}$$ is continuous using the universal property of quotient spaces. Let ##f:[0,1]→X## and ##g:[1,2]→Y## f and y are continuous, thus...
31. ### I Proving a function f is continuous given A U B = X

Basically with this problem, I need to show that f is continuous if A and B are open and if A and B are closed. My initial thoughts are that in the first case X must be open since unions of open sets are open. My question is that am I allowed to assume open sets exist in Y? Because then I can...
32. ### I Is time continuous or discrete in quantum physics?

I was working on plotting fidelity with time for two quantum states. First I used discrete time( t= 0,1,2,3...etc) to plot my fidelity. I got constant fidelity as 1 with continuous value of time. Next I used discrete set of values ( t=0 °,30 °,60 °,90 °). Here I saw my fidelity decreases and...
33. ### B Torque applied by a continuous mass instead of point particle

I came across this 'problem' when I was trying to think about how a torsion spring would apply torque in something like a miniature catapult. I understand that in the context of something like turning a wrench, we can find the net torque on the wrench by treating the hand applying the force as...
34. ### Showing continuous function has min or max using Cauchy limit def.

Problem: Let ## f: \Bbb R \to \Bbb R ## be continuous. It is known that ## \lim_{x \to \infty } f(x) = \lim_{x \to -\infty } f(x) = l \in R \cup \{ \pm \infty \} ##. Prove that ## f ## gets maximum or minimum on ## \Bbb R ##. Proof: First we'll regard the case ## l = \infty ## ( the case...
35. ### Probability: Prior Distribution for a Continuous Variable

Hi, I was attempting this problem from the MIT OCW website probability and statistics course. Context: When Jane started class, she warned Jon that she tends to run late. Not just late, but uniformly late. That is, Jane will be late to a random class by ##X## hours, uniformly distributed over...
36. ### The maximum efficiency of two continuous processes

I think it reaches its maximum efficiency when it is two continuous Carnot process. Its efficiency then will be H= 1-(W1+W2)/(Q1+Q2), with W1/Q1>=T2/T1 and W2/Q2>=T3/T2 therefore H<= 1- (Q1.T2/T1+Q2.T3/T2)/(Q1+Q2), that is as far as i can go, have not got a result yet
37. ### B Are continuous functions on sequentially compact sets u-continuous?

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...

46. ### Is the given function continuous at x= 0? f(x) = {sin(1/x) if x≠0, 1

I know that the function, g(x)= sin(1/x) has infinite oscillations when the values of x get closer and closer to 0. So its limit does not exist (from graphing it). However, the way that we defined f(x), at x=0, f(x)=1, but f(x)= sin(1/x) on (0,infinity). I have an issue in general showing that...
47. ### Photoelectric effect and continuous energy function

E=hf-W where W is a work function. However we know that electrons in an atom will be excited only when radiated with photons of n*f0 discrete number of frequencies. where E=hf-W is a continuous function. Is this because energy level is continuous within a conductor? If we think of only...
48. ### Calculating a mean related to a continuous random variable

I am not sure about how to approach this. Since the volume is uniformly distributed, the mean volume is ##(5.7+5.1)/2=5.4##, which is less than ##5.5##. From this, I could say that, on average, the producer won't spend any extra dollars. But then I thought that maybe I should interpret this as...
49. ### I Continuous vs discrete Universe

I've come across the question of continuity vs discreteness in different articles, discussions, etc. but I'm not sure that I am 100% clear on what the precise question is. My basic interpretation of it is a question of whether the Universe is made up of lots of separate entities which all...
50. ### Electric Field and Continuous Charge Distribution

I sort of understand the meaning of this integral, but I don't know how to evaluate it. I have never evaluated a volume integral. It would be very helpful if someone could explain in other words what this integral means and give an example evaluating it. This is from Purcell's Electricity and...