Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) ...

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twice differentiable functions

Let $f:\mathbb R \to \mathbb R$ be a twice continuously differentiable function, with $f(0)=f(1)=f'(0) = 0$. Then $f^{"}$ is the zero function. $f^{"}(0)$ is zero. $f^{"}(x)=0$ for some $x \in $ ...
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Definition of limit as $x\rightarrow \infty$

Every time i get confused with the definition of $\lim_{x\rightarrow \infty}f(x)=L$. I could not find a reference that will give the definition. I am trying to write what i understood. See if this is ...
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Polynomials and Lipschitz function

Let $f(x) = x^4 + 11x^2 + 9x -5$ and let $M > 0$. Show that f is a Lipschitz function on the interval $[-M, M]$ I honestly cannot figure out how to start this proof. Nothing similiar is in the ...
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$f$ is continuous, is $1/f$ continuous

Let $f: A \rightarrow \Bbb{R}$ be uniformly continuous. Suppose there exist $k>0$ s.t. $|f(x)| \ge k$ for all $x \in A$. Show that the function $1/f$ is also uniformly continuous on $A$. My proof ...
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Nonuniform Continuity Criteria with cos(1/x)

Use the nonuniform continuity criteron to show that $f(x) = cos\left(\frac{1}{x}\right)$ is not uniformly continuous on $(0, \infty)$ My proof: Let $(x_n)$ be defined by $x_n = \frac{1}{2n\pi}$ ...
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Continuous Functions and neighborhood

Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be continuous and let $\beta \in \Bbb{R}$. Show that if $x_0 \in \Bbb{R}$ is such that $f(x_0) < \beta$, then there exist a $\delta$ - neighborhood $V$ of $x_0$ ...
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Closed graph of a function

I have difficulties to answer at that question: Let $X$ be a Hausdorff and compact topological space, and let $Y$ be a topological space. Let $f:X→Y$ be such that $G(f) = \{(x,f(x))|x∈X\} $ is a ...
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31 views

Given a Lipschitz continuous function, prove there is some $M > 0$ such that $|f'(x)| < M$

Given $f: I \rightarrow \mathbb{R}$ is Lipschitz continuous, where $I \in \mathbb{R}$ is an interval, prove there exists some $M > 0$ such that if $f$ is differentiable at a point $x \in (a,b)$, ...
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If $f$ is defined on $R$ and $f(K)$ is compact whenever $K$ is compact, then is $f$ continuous on $[a,b]$?

If $f$ is defined on $R$ and $f(K)$ is compact whenever $K$ is compact, then is $f$ continuous on $[a,b]$? I know that if $f : K → R$ is continuous and $K \in R$ is compact, then $f(K)$ is compact, ...
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A map between metric spaces preserving convergent sequences is continuous

Pugh, "Mathematical Analysis", exercise 2.17: Assume $f : M \to N$ is a map from one metric space to another which satisfies the following condition: for every convergent sequence $(a_n) \subset ...
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Is this (set-)difference of spaces of functions nonempty?

Let $g_y$ be a family of continuous functions on $[a,b]$, indexed by $y\in [0,1]$, such that $(x,y)\to g_y(x)$ is also continuous. Denote the set of all such families that additionally fulfill ...
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Binary operations being continuous under a topology?

For a set $S$ and some function $f: S \rightarrow S$ and $a\in S$, $f$ is continuous at $a$ under a topology $N$ if for all neighborhoods $N_1(f(a))$ there exists a neighborhood $N_2(a)$ such that ...
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Show that $\rho : X \times X \mapsto \mathbb{R} $ is continuous function on $(X \times X, \tau)$

Show that $\rho : X \times X \mapsto \mathbb{R} $ is continuos function on $(X \times X, \tau)$ where $\tau ((x_1,x_2), (y_1,y_2)) = \sqrt{\rho (x_1-y_1)^2 + \rho (x_2-y_2)^2}$ and $X \times X$ is the ...
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Prove $f(x)=\|x\|$ differentiable everywhere but in $\{0\}$

I have the function $f: \mathbb R ^n \to \mathbb R$ where $f(x)=\|x\|$. I have to prove that $f$ is differentiable on $E$, where $E=\mathbb R^n \setminus \{0\} $, and show its derivative (for $x \ne 0 ...
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‘Every continuous real-valued function on $X$ achieves a minimum’ is a topological property.

Suppose that a topological space $X$ has the property that every continuous real-valued function on $X$ takes on a minimum value. I need to show that any topological space that is homeomorphic to $X$ ...
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examples of continuous function, open set

(a) Give an example of a continuous function and an open set U such that $f(U)$ is not open. (b) Give an example of a continuous function and a set U, which is not open, such that $f^{-1}(U)$ is ...
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32 views

How do I show that a function is continuous in topology

Let $(X,T_1)$ and $(Y,T_2)$ be topological spaces and $z\in Y$ and define $f(x) = z$ for all $x\in X$ How do I show that $f(x)$ is continuous? $f(x)$ is continuous at x at if $f(x) = y \in O$ where ...
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A question regarding the definition of differentiable.

Let $f:\mathbb R\to\mathbb R$ be a differentiable function, then it holds that $$f(y)=f(x)+f'(x)(y-x)+o(|y-x|).$$ Here a function $g(x)=o(|x-y|)$ if $$\lim_{x\to y}\frac{|g(x)|}{|x-y|}=0.$$ If we let ...
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“Faint” continuity

Definition: A function $f:\mathbb{R}\to \mathbb{R}$ is called faintly continuous in $x$ if there are two series $x_n < x < y_n$ with $\lim_{x_n \to x} f(x_n) = \lim_{y_n \to x} f(y_n) = f(x)$. ...
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Prove every continuous function f: M -> R is a constant function

Assume M has only a countable or finite number of points and M is connected. Prove that every continuous function f:M->R is a constant function on all of M. Here is what I have so far: If f: M->R ...
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continuous functions on unit circle — group isomorphism

Let $G$ be the additive group of continuous real-valued functions on the unit circle $S^{1}$, let $H$ be its subgroup of $\mathbb{Z}$-valued functions, and let $\tilde{G}$ be the additive group of ...
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Is this function Hölder continuous?

Define $f:[0,1]\rightarrow \mathbb{R}$ as $$f(x)=x^\alpha \int_x^1 y^{-\alpha-1}(y-x)^{-\alpha}dy, \quad x\in [0,1],$$ where $\alpha\in (0,1/2)$ is some fixed parameter. Is $f$ Hölder continuous of ...
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For any arbitrary $\mu$ there is an arbitrary interval $[x,x+h]$ where $\left| f(x+h) - f(x) \right| < \mu h$

In Royden's textbook, this is taken for granted in the proof that if $f$ is absolutely continuous and $f'(x) = 0$ almost everywhere, then $f$ is constant. He says the following We aim to show that ...
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Exercise #9 in chapter 11 of Rudin's Principles of Mathematical Analysis.

Suppose $f$ is Lebesgue integrable on $[a,b]$. Let $F(x)$=$\int_{a}^x fdt$. Then prove that $F$ is continuous on $[a,b]$. I know that $F$ is continuous almost everywhere, because $F'(x)=f(x)$ ...
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31 views

Discontinuous/Continuous Function

I am having a hard time coming up with a function for this question. Any ideas? A function that is discontinuous at even natural numbers and continuous everywhere else.
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Proving that function continuous has a maximum value when function has few properties

Question: Let $f: \Bbb{R} \rightarrow \Bbb{R}$ continuous function, which has following properties: $\lim_{x\to -\infty}f(x)=0$ $f(0)=2$ decreasing when, $x\ge2 $ Prove that function has upper ...
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How can I prove the equation has unique positive real solution?

Without using derivative, prove that the equation $$x^5-2x^4-3x^3-4x^2-5x-6=0$$ has unique positive real solution. I tried, consider function $f: \mathbb{R} \rightarrow \mathbb{R}$ with ...
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28 views

Using definition of continuity to show that a function is continuous at the point a.

I am having some trouble with this particular type of question which asks, Use the definition of continuity and the properties of limits to show that the function is continuous on the given ...
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Improper integral proof: limit of integral exists when the integral is continuous?

We're trying to prove the integral $$\int_0^1\frac{\cos x}{x^\frac12}\,dx$$ exists as an improper integral. My teacher says that in order to prove there exists the limit of $\int_a^1\frac{\cos ...
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If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$

Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Show that if $f$ is continuous, then $f(x) = cx$ for all $x \in \mathbb{R}.$ I find it ...
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Why is $x\in X$ a weak star continuous linear functional in the dual?

I am reading an excerpt from Infinite Dimensional Analysis by Aliprantis and on page 235 it claims that if $X$ is a normed space, then "$x$ is a weak* continuous linear functional by definition". ...
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Is the following function Lipschitz continuous, uniformly continuous, or neither?

$f: x \mapsto ax + b$ on $\mathbb{R}$ ($a,b\in\mathbb{R}$) $f: x \mapsto x^2$ on $(0, 1)$
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Given a function $f: X \to Y$, if $X$ is compact, prove the graph $g = (x, f(x))$ is compact in $X\times Y$

Given a function $f: X \to Y$, and graph of $f, g = \{(x, f(x)): x\in X\}$ in metric space $X\times Y$ (a) Suppose that $X$ is compact. Prove that $f$ is continuous if and only if $g$ is a compact ...
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Least involved proof for continuos functions => uniform continuos functions on [a,b]

I have been looking at this proof in my textbook and seem to always get lost in its logic, its roughly 3 pages long. The proof is: If f is continuous on a closed interval [a,b], then f is uniformly ...
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Cauchy Functional Equation Extension

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y),\forall x,y\in\mathbb{R}$, $\lim_\limits{x\to +\infty}{f(x)}=+\infty$ and $f(e^x)=e^{f(x)}, \forall x\in (0,+\infty)$ ...
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41 views

Prove there is not continuous surjection f: Sn --> Rn where Sn is a sequence in Rn+1

Let $S^n = \{(x_1, . . . , x_{n+1}) \in \mathbb{R}^{n+1} : \sum_{k=1}^{n+1} x_k^2 = 1\}$. Prove that there is no continuous surjection $f : S^n \to \mathbb{R}^n$.
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Continuity of multivariable piecewise function (sin, cos)

Let $$ f(x,y) = \begin{cases} \dfrac{\ (x-2y)\sin(xy)}{(xy^2 + x)},& y\ne0\wedge xy\ne-1\\ x^2,& y=0\vee xy=-1. \end{cases} $$ I should analyze continuity in each domain point and justify ...
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Continuity of function defined on R to Q [duplicate]

Let $f:[1,3]\to\mathbb{Q}$ be a continuous function such that $f(2)=10$. Then $f(1.8)$ equals a) 1 b) 5 c) 10 d) 20.
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Continuity of a function.

If derivative of function is Infinite or not defined at some point them can fuction be continuous at that point. Please make it clear by proper example.
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Proof concerning a function continuous in an interval.

I have been working on a question from Richard A. Silverman's Modern Calculus and Analytic Geometry which would appear quite simple except one case prevents me from making more progress. The question ...
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1answer
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How to approximate the derivative of a stock price over time?

My high school marketing class is about to do a unit on stocks. We're going to make "pretend" investments over the next month or so, and have a competition to see who has the highest gains. These are ...
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Why weren't continuous functions defined as Darboux functions?

When we were in primary school, teachers showed us graphs of 'continuous' functions and said something like "Continuous functions are those you can draw without lifting your pen" With this in ...
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If $f: \Bbb R\rightarrow \Bbb R $ is monotonic and onto, prove that $f$ is continuous.

If $f: \Bbb R\rightarrow \Bbb R $ is monotonic and onto, prove that $f$ is continuous. (Hint: given any $x \in \Bbb R$ and any $\epsilon \gt 0$, there is $x_1$ with $f(x_1)= f(x) - \epsilon$ and ...
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Superior and inferior limits with continuous functions

Given two continuous functions $f$ and $g$, from $\mathbb R$ to $\mathbb R$ and with $f \leqslant g$. If we have a sequence $a_n$ such that it has a limit $a$. Is it true that lim sup $(f(a_n)$, ...
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1answer
36 views

Continuity of multivariable piecewise function (cos, sin)

Continuity of multivariable piecewise function (sin, cos) Let $$ f(x,y) = \begin{cases} \dfrac{\cos(x)xy⁴ + a\sin(x⁴)}{(x^2 + y^2)}\quad& \text{if}\quad (x,y)\neq(0,0)\\ 0\quad& \text{if} ...
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Question about limit points with continuous maps. What have I done wrong?

I'm trying to do this exercise: Suppose that $f:X \rightarrow Y$ is continuous. If $x$ is a limit point of the subset $A$ of $X$, is it necessarily true that $f(x)$ is a limit point of $f(A)$? ...
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Continuity of solutions for systems of rational ODEs

If you have a system of ODEs of the from $\frac{d\mathbf{x}}{dt}=\mathbf{f(x,p)}$ where $\mathbf{x}$ is a vector valued variable, and $\mathbf{p}$ is a vector of parameters, and $\mathbf{f(x,p)}$ is a ...
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2answers
45 views

If $f(x)$ is discontinuous at $0$, does that mean ${f(x)}^3$ is discontinuous at $0$?

If $f(x)$ is discontinuous at $0$, does that mean ${f(x)}^3$ is discontinuous at $0$? I know this holds true for the function $f(x)= \frac{1}{x}$, since $\frac{1}{x^3}$ is discontinuous at $0$ as ...
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Suppose $f:E \rightarrow \mathbb{R}$ is continuous at $p$. Prove that if $f(p) > 0$, then there is $\delta>0$ s.t. $f(x) \geq f(p)/2$.

Suppose $f:E \rightarrow \mathbb{R}$ is continuous at $p$. Prove that if $f(p) > 0$, then there is $\delta>0$ s.t. $f(x) \geq f(p)/2$ $\forall x$ in $E$ s.t. $|x-p| \leq \delta$. I couldn't ...
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1answer
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Open Maps between topological spaces

Let $X,Y$ be topological spaces. A map $f$ from $X$ to $Y$ is open if for every open set $U$ in $X$ , $f(U)$ is open in $Y$. Let $X= \mathbb{R}$ and $Y= \mathbb{S^1}$ and $f: t \rightarrow e^{2\pi i ...