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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continuous (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 $\phi:G\rightarrow Sym(M)$, where $Sym(M)$ is the symmetry group of M, ...
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If $x,y \in \bar{\mathbb{R}}$ then is $g(x,y)=xy$ continuous?

Suppose we assume the convention that $0 \cdot \infty =0$. If $\bar{\mathbb{R}}$ is the extended real line and $x,y \in \bar{\mathbb{R}}$, then is $g(x,y)=xy$ continuous? Why? I do not think it is as ...
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Being Careful with open sets

I have a function $f$ on [$0,1$]. I don't know whether on not it will be continuous at $1$ but I know if you pick any $0<\eta<1$ then it will be continuous on [$0,\eta$]. Surely this is enough ...
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Continuity of convex functions that have continuous restrictions to closed subspaces

Let $X$ be an infinite-dimensional normed vector space , let $U\subset X$ be an infinite-dimensional closed subspace, and let $f:X\to[0,\infty)$ be convex. Question: If the restriction $f|_U$ is ...
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Show that exist open ball B, such that $f(B)\cap g(B)=\emptyset$

$M,N$ metric space. Let $f,g:M\to N$ continuous in a point $a\in M$. If $f(a)\neq g(a)$, then exist a open ball B of center a, such that $f(B)\cap g(B)=\emptyset$. In particular, if $x\in B$, then ...
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Is the function $g:\mathbb{R} \setminus\{0\} \to \mathbb{R}$ given by $g(x) = 1/x^3$ continuous? Why or why not?

Is the function $g:\mathbb{R} \setminus\{0\} \to \mathbb{R}$ given by $g(x) = 1/x^3$ continuous? Why or why not? A real valued function $f$ is continuous at $a \in \mathbb R$ if the $\lim_{x ...
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Analysis Proof- different conditions.

A continuous function on $[a,b]$ is also uniformly continuous on $[a,b]$. The following tries to illustrate what happens when the interval is not closed: Show: $f(x) = \frac{1}{x} $ is not ...
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Prove using the ε-δ definition that f(x) is continuous at the following points? [closed]

How can I use the $\epsilon$-$\delta$ definition to do this for $f(x) = \frac{x + 2}{x^2 + x + 1}$ at $x=1$ and for $f(x) = \log(x^2 + 1)$ at $x=0$? I can do it for a normal function but I get ...
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Prime ideals in $C(X)$ and $C^*(X)$ and to be correspond

we know that every maximal ideal in $C(X)$ is in this form: $$M^p=\left\{\,f \in C^*(x):\ p\in cl_{\beta X} Z\left(f\right)\,\right\}$$ and every maximal ideal in $C^*(X)$ is ...
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Let $f$ be strictly increasing and $g,\ g\circ f$ is continuous. Does this implies that $f$ is continuous?

Let $f,\ g: \mathbb R \to \mathbb R $ be two nonconstant functions. Let $f$ be strictly increasing and $g,\ g\circ f$ is continuous. Does this implies that $f$ is continuous? How to think this ...
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42 views

Prove continuity of a function that is defined through a geometric construction

I need to prove that a function is continuous, but it is not defined explicitly,it's like this: given a point $P$ on a circumference and an angle $0\le a\le \frac{\pi}{2}$ defined by $P$ and another ...
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34 views

Does the sequence have a uniform limit? How do I show this?

I am having problems with the following exercise, I already did part $(i)$ and $(ii)$, I am having problems with $(iii)$. Exercise: Let $ \large f_n(x)=\left\{ \begin{array}{ll} ...
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28 views

Is every continuous function from co-finite topological space to a $T_1$ space is constant?

Let $X$ be a co-finite topological space and $Y$ a $T_1$ space. Is it true whether every continuous function from $X$ to $Y$ is constant? I cannot prove it. I cannot find a counterexample.
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How to show that this function is continuous

I'm trying to show continuity of the function $$\frac {\ln(1+x^2+y^2)}{x^2+y^2}$$ for $(x,y)\neq 0$, $$f(x,y)=1$$ for $(x,y)= 0$, on $\mathbb{R}^2$ But I am not able to. The numerator is stuck for me ...
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26 views

Given that the function is of class $C^2$ prove the following.

Let $g:\mathbb{R} \to \mathbb{R}$ be of class $C^2$. Show that $$\lim_{h \rightarrow 0} \frac{g(a+h)-2g(a) +g(a-h)}{h^2} = g''(a)$$ How should one approach such questions? There are so many things ...
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Show that the function is of class $C^1$

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ such that $f(0,0)=0$ and $f(x,y)= \frac{xy(x^2-y^2)}{(x^2+y^2)}$, if $(x,y) \neq (0,0)$. Show that $f$ is of class $C^1(\mathbb{R}^2)$. If we use the ...
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How is the epsilon-delta definition of continuity equivalent to the following statement?

Claim: A function $f: \mathbb{X} \to \mathbb{Y}$ is continuous if given any open set $\mathbb{U} \subseteq \mathbb{Y}$ the inverse image $f^{-1} (\mathbb{U}) \subseteq \mathbb{X}$ is open. How is ...
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48 views

Continuously Differentiable in $\mathbb{R^2}$

I understand the concept of continously differentiable (first derivative is continuous) in $\mathbb{R}$, however what does it mean for the RHS of: $\dfrac{d}{dt} ...
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51 views

Continuous Function on a closed interval $[a,b]$

A continuous function on a closed interval $[a,b]$ is bounded and attains its bounds. A continuous function on $(a,b]$, $[a,b)$, or $(a,b)$ may not be bounded at all or attain its bounds, this fact ...
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Is there a 2D analogue of Thomae's function?

Thomae's function, $$ f(x)= \begin{cases} 0 & \text{$x$ is irrational}\\ \frac{1}{p} & \text{$x = \frac{p}{q}$ where $\gcd(p,q) = 1$} \end{cases}, $$ is an example of a function from ...
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Point about the theorem and proof of the inner product being a continuous function.

In moving to show that the inner product, $\langle\cdot,\cdot\rangle$ is a continuous function I have the following theorem in my notes (also on page 59 of "Linear Functional Analysis", Rynne and ...
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Prove/disprove if $f$ is continuous on $[0,1]$, and absolutely continuous on $(a,1], a\in (0,1)$, $f$ is absolutely continuous on $[0,1]$.

Problem statement: Suppose $f$ is a real-valued, continuous function on $[0,1]$, and $f$ is absolutely continuous on $(a,1]$ for every $a \in (0,1)$. Is $f$ necessarily absolutely continuous on ...
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Various definitions of “topological immersion”

In Spivak's book on differential geometry he defines a topological immersion $f$ as "$f$ is a continuous function that is locally one-one". In my limited experience with the category $\mathsf{Top}$, ...
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Understanding of the proof of “intermediate value thm”

Theorem. Let $f$ be a continuous function from $[a,b] \rightarrow \mathbb{R}$, and $f(a) \not = f(b)$. Then for all C between $f(a)$ and $f(b)$, there exists some point $c$ in $[a,b]$ such ...
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modulus of continuity

the Hölder $\rho$-continuity is defined as $$|f(x)-f(y)| \leq K_1 |x-y|^\rho.$$ I'm doing a research problem right now and might need the following condition $$K_2 |x-y|^\rho\leq|f(x)-f(y)| \leq K_1 ...
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Restricted function continuity

Given the function: $$f(x) = \begin{cases} \frac{x^2y}{x^4+y^2} &\mbox{if } (x,y) \ne (0,0) \\ 0 & \mbox{if } (x,y)=(0,0). \end{cases} $$ Show that function restricted to random line passing ...
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Show that the function $x\mapsto x\sin x$ is continuous at $x = 0$

Use the definition of continuity to show that the function: $f: \mathbb R \to\mathbb R$ defined by $f(x) = x\sin x$ is continuous at $x = 0$. Def: A real valued function $f$ is called ...
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Prove using algebra of continuous functions that $f$ is continuous in $\mathbb{R}$

Let us consider $f : \mathbb{R} \to \mathbb{R}$ defined by $$f(x) =\begin{cases} x^2 \sin \frac{1}{x},& x \neq 0\\ 0,& x = 0\end{cases}.$$ By using algebra of continuous functions ...
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Show that if $f(x_0) \neq 0$, then there is an interval where $f(x) \neq 0$ if $f$ is piecewise continuous and $f(x) = \frac{1}{2} [f(x-) + f(x+)] $

Suppose $f$ is piecewise continuous on $(a,b)$ and $f(x) = \frac12 [ f(x-) + f(x+) ]$, where $f(x+)$ and $f(x-)$ are the right and left limits, respectively. Show that if $f(x_0) \neq 0$ for some $x_0 ...
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Continuity of anti-derivative multivariate functions

I just need some hints to solve the following problem. Let $f(x_1,\ldots, x_n)$ be a continuous and integrable function on $\mathbb{R}^n.$ Is the function $$ g(x_1,\ldots, x_{n-1}) = \int ...
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Differentiability and continuity of a multivariable function

Let $f:\mathbb R^2\to \mathbb R$ be defined by $$f(x,y)=\begin{cases}\frac{x|y|}{\sqrt{x^2+y^2}},& (x,y)\ne(0,0)\\ 0,& (x,y)=(0,0).\end{cases}$$ For which non-zero vectors $u$ does ...
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63 views

Alternative to dense subsets for non-Hausdorff spaces

Dense subsets of a topological space $X$ satisfying strong enough separation axioms (Hausdorff is enough) have the property that for any two continuous maps $f:X \to Y$ and $g:X \to Y$ into any ...
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Prove that continuous functions f and g intersect

Let $f,g$ be continuous functions on $[a,b]$ with $f(a) \geq g(a)$ and $f(b) \leq g(b)$. Prove there is some $x$ in $[a,b]$ where $f(x)=g(x)$. My question is about my following method and not others ...
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Continuity of the composition of continuous functions

I have a question about the continuity of composite functions. Let $f:[a,b] \rightarrow \mathbb{R}$ and $g:[c,d] \rightarrow [0,1]$ be continuous functions. Define the function $h:[a,b]\times [c,d] ...
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Rudin 4.22 Let $f$ be a continuous function from $X$ to $Y$ (metric spaces). If $E$ is connected, then $f(E)$ is connected

Rudin 4.22. Let $f$ be a continuous function from $X$ to $Y$ (metric spaces). If $E$ is connected, then $f(E)$ is connected. Could someone check this proof: Proof: I will show the ...
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Is this function strongly convex?

Let A,B be two intervals in $R$ and let $f(x,y):A\times B\rightarrow R $ be a continues function. Assume that $f$ is convex in both $x$ and $y$ Is the following function $g:A \rightarrow R$ is ...
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If $f:\mathbb R^n\to\mathbb R$ is twice continuously differentiable, then $\nabla f$ is Lipschitz continuous

Let $f\in C^2(\mathbb R^n)$. I've read that since $f$ is twice continuously differentiable, $\nabla f$ is Lipschitz continuous. Is that really true? By the mean-value theorem, $$\left\|\nabla ...
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A Map From $S^n\to D^n/\sim$ is Continuous.

The following question is motivated by Thomas's answer here which can be used to prove that $\mathbf RP^n$ is same as the space obtained by identifying the antipodal points on the boundary circle of ...
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Characterizing the continuous functions from $\mathbb{N}$ with the cofinite topology to $\mathbb{R}$

Let $\mathbb{N}$ have the co-finite topology, and let $\mathbb{R}$ have the usual topology. Then what functions from $\mathbb{N}$ to $\mathbb{R}$ are continuous? I think the constant functions would ...
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Show $\lim_{m \to \infty ,n \to \infty } f(\frac{{\left\lfloor {mx} \right\rfloor }}{m},\frac{{\left\lfloor {ny} \right\rfloor }}{n}) = f(x,y)$

The question is Suppose $f(x,y)$ is defined on $[0,1]\times[0,1]$ and continuous on each dimension, i.e. $f(x,y_0)$ is continuous with respect to $x$ when fixing $y=y_0\in [0,1]$ and $f(x_0,y)$ is ...
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Continuity ( Functions of 2 variables ).

Given , $$ f(x,y) = \begin{cases} \dfrac{xy^{3}}{x^{2}+y^{6}} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \\ \end{cases} $$ We need to check whether the function is continuous at ...
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Extending continuous function from a dense set

If $X$ is a metric space and $Y$ a complete metric space. Let $A$ be a dense subset of $X$. If there is a uniformly continuous function $f$ from $A$ to $Y$, it can be uniquely extended to a uniformly ...
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31 views

Increasing and decreasing piecewise function on an interval

I'm working on a problem that involves finding the intervals where a function $f$ is increasing and decreasing. Given the function$$ f(x) = \cases{ x+7 & \text{if } x\lt -3\cr |x+1| & ...
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If $\ne: X \times X \to S$ is continuous, is X hausdorff?

The Sierpiński space is defined like so: $$S = (\{\top, \bot\}, \{\emptyset, \{\top\}, \{\top, \bot\}\})$$ (A nice way to visualize is to take [0, 1], and glue 0 on $\bot$ and (0,1] onto $\top$.) ...
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40 views

Using $\epsilon$-$\delta$ argument to show continuity

Show using the $\epsilon$-$\delta$ definition of continuity that $f(x)=\begin{cases} 11&\text{if}~0\leq x\leq 1\\x&\text{if}~ 1<x\leq 2\end{cases}$ is continuous on $[0,1)\cup (1,2]$ ...
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Exercise of Application of Implicit Function Theorem

Let $f\colon U \subset \Bbb R^2 \to\Bbb R$ such that $$\forall (x,y) \in U \quad (x^2 + y^4)f(x,y) + (f(x,y))^3 = 1$$ Prove that $f$ is $C^\infty$.
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If a function $f: X \to Y $ $X, Y $ are some normed spaces, is continuous, then $\|f(x)\|< \infty , x \in X$ and why? [closed]

If a function $f: X \to Y $ $X, Y $ are some normed spaces, is continuous. Then is $\|f(x)\|< \infty , x \in X$ and why? I am thinking yes on this one (because otherwise, it would not be ...
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52 views

Continuity of the inverse map

If we have a function $F(x): \mathbb{R^4} \rightarrow \mathbb{R^3}$. Defined as \begin{align} x_1\, x_4&=y_1 \\ x_2\, x_4&=y_2 \\ x_1^2+x_2^2-x_3^2&=y_3 \end{align} Can a continuous ...
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Counter example of non continuity

I present in the following a variation of the problem described in Continuity of a deterministic function generated from a probability function. There, it has been proved that $g(x)$ is not ...
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Non-periodic continuous functions takes maximum and minimum at every compact subset of the domain?

Let $f:\mathbb R \rightarrow \mathbb R$ a continuous and periodic function, with period $p$. By Weierstrass's theorem, the restriction $f|_{[x_0,x_0+p]}$ is bounded and takes its maximum and minimum ...