Questions tagged [continuity]
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) or (general-topology)
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Applying Fundamental Theorem of Calculus to improper integral
Let's consider $f:~]-\infty,\infty[~\to\mathbb{R}$ a continuous function. Our professor nonchalantly said that if we assume that $\int\limits_{-\infty}^xf(t)dt$ exists, then differentiating by using ...
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$B\subset\mathbb{R}$ and $f:B\to\mathbb{R}$ is an increasing function. $f$ is continuous at every element of $B$ except for a countable subset of $B$.
I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The following exercise is Exercise 22 on p.39 in Exercises 2B in this book.
Exercise 22
Suppose $B\subset\mathbb{R}$...
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If s and t are positive step functions defined on an interval [a,b], is s/t also a step function?
I think it is, assuming that both functions are defined on the same partition (joint refinement), if we let s = $\sum_{i=1}^n c_i1_{[p_{i-1},p_i]}$ and t =$\sum_{i=1}^n d_i1_{[p_{i-1},p_i]}$ then $\...
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Does uniform convergence on any interval of form $[-M,M]$ imply continuïty of limit function?
For the past few days I've been studying Analysis and more specifcally the topic of continuïty. I was making some exercices on this topic and got stuck with a problem which goes as follows. Let $(f_n)...
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Is $f(x)=x^{2}$ a continuous function defined on the following topological spaces?
I have the following statement which I need to prove or disprove.
Suppose $f(x)=x^{2}$ from $(\mathbb{R}, \tau_{5}) \rightarrow (\mathbb{R}, \tau_{cofi})$, where $\tau_{5}=${$A\subseteq \mathbb{R}:5\...
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Image of compact interval under continuous function
We know the image of a compact interval under continuous function is compact interval.
My doubt is, suppose a continuous function on closed and bounded interval $[a,b]$ is costant.
Then images reduces ...
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Need help understanding step functions notations and concepts
I am confused on the notations of step functions.
If we have a sequence of step functions $\{s_n : [0,1] \rightarrow \mathbb R \}_{n = 1}^{\infty}$ , what would $s_n(0)$ look like, as in if I plug ...
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$\|L\|\leq M<\infty\implies L$ is continuous.
I'm trying to prove that in a normed vector space $E$, a linear functional $L\colon E\to \mathbb{R}$ is continuous if
$$\exists M\in \mathbb{R},\; \|L\|<M.$$
My attempt: Let $x\in E$ and $\...
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Continuity of this two variable function with inequalities
I would like to know if I am wrong with the following:
Prove or disprove the continuity of the function
$$f(x, y) = \begin{cases} \frac{4x^3y^3(y^4-x^8)}{(x^8+y^4)^2} & (x, y) \neq (0, 0) \\\\ 0 &...
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Some help in proving differentiability at the origin.
How can I prove the function is continuos at the origin with the distance majorization? Here is what I tried, but I end up in a dead end.
$$f(x, y) = \frac{xy^2}{(x^2+y^2)\sqrt{x^2+y^2}}$$ when $(x, y)...
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Are inf and sup continuous functionals in general?
Let $X$ be any topological space and $\bar{\mathbb{R}} = [-\infty, \infty]$ with the standard topology. Is it true, in general, that the functionals $$\inf: C(X,\bar{\mathbb{R}}) \to \bar{\mathbb{R}}, ...
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If a (partial) derivative is continous near x, does that imply that the (partial) derivative at x is either continous or non-existent?
Let $x \in \mathbb{R}^n$, and $A$ be a neighborhood of $x$. Let $f$ be a function with a (partial) derivative that exists and is continous on $A-\{x\}$. Does this imply that the (partial) derivative ...
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Logical implication in existence of partial derivatives with non differentiable function
I ask for some help in unknotting this chain of reasonings, in particular in spotting logical errors due to wrong implications and possible mathematical misbeliefs about this problem.
Theorem: If $f(x,...
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Is there a simple way to interpolate smoothly between levels of a complex-valued quadratic map?
I have two complex numbers, $a = x_1 + y_1 i$ and $c = x_2 + y_2 i$. These serve as inputs to a quadratic map $f_n = f_{n - 1}^2 + c$, with $f_0 = a$. Thus the first few iterations of the map are:
$...
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What is the type of discontinuity of $e^{\frac{1}{x}}$ at zero?
The limits of this functions at zero are:
$\lim_{x \to 0^+} e^{\frac{1}{x}} = \infty $ an infinity discontinuity
$\lim_{x \to 0^-} e^{\frac{1}{x}} = 0 $ a removable discontinuity
The question is: Is ...
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Show if $f:\mathbb{Z}_p\to\mathbb{Q}_p$ is continuous such that $f(n)=(-1)^n$, then $p=2$.
Let $\mathbb{Q}_p$ denote the p-adic rationals and same for the integers. Suppose $f:\mathbb{Z}_p\to\mathbb{Q}_p$ is continuous, with the additional property that at $n\in\mathbb{Z}^{\geq 0}$, $f(n)=(-...
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How might one describe the continuity of a function with pure mathematical language?
In high school calculus, we often need to reference that a function is continuous in order for it to be differentiable. Is there a way I can write this in pure set theory or order theory? It would ...
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Finding norm of integral operator on C[0,1] [duplicate]
I need help with the following:
Find norm of inegral operator $K:C[0,1] \to C[0,1]$ which is defined as
$$Kf(x) = \int_0^1 k(x,y)f(y)dy, \forall f \in C[0,1], x \in [0,1],$$
where $k \in C[0,1]^2$.
...
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Uniform convergence implies convergence of suprema?
Let $f_n:M \rightarrow \mathbb{R}, M \subset \mathbb{R}, n\in \mathbb{N}$ be a sequence of continuous functions that converges uniformly on $M$ to a continuous function $f:M \rightarrow \mathbb{R}$. ...
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Multivariable calculus: is this exercise a mess?
I have to find the points at which the following function is continuous, has partial derivatives and it's differentiable.
$$f(x, y) = \begin{cases} x & y < x^3 \\\\ y & y \geq x^3 \end{...
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Exercise on multivariable calculus: am I reasoning right?
Consider
$$f(x, y) = \begin{cases} x^2+y^2 & x^2+y^2 \geq 1 \\\\ \frac{1}{x^2+y^2} & 0 < x^2+y^2 < 1\end{cases}
$$
Now, the domain of $f$ is: $D: \mathbb{R}^2\backslash\{(0, 0)\}$
When ...
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Continuity of optimal value of an optimization problem with respect to a parameter in constraint [closed]
I have some optimization problem of the following form
min h(x)
subject to g(x) <= D
where h is a convex function. And I wish to prove that f(D) = min h(x) is continuous. What are the possible ways ...
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Proving hemicontinuity.
In my Mathematics for Economics course, we have started studying hemicontinuity. The concept is very clear to me, the problem is that I cannot understand how to prove it. Here is an example, from the ...
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if second derivative of f exist at one point can we say its first deirvative is continuous at its neighborhood [closed]
there is function that can be differentiable only at one point ,but if we only kown that the second derivative of f exists at x=0 ,
can we say the first derivative of f is continuous in the small ...
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Continuous transformations that rearrange the coordinates of points [closed]
What are continuous transformations $F:\mathbb{R}^n \to \mathbb{R}^n$ permuting the coordinates of points $x=(x_1, \ldots, x_n)$? Are they permutations of coordinates of points and therefore linear ...
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A question on on inverse image of mapping
Given $F:[a,b]\to \mathbb{R }$ to be continuous function.
Suppose $F(I)$ and $F(I')$ are non-overlapping closed intervals (for some closed intervals $I, I' \subseteq [a,b]$).
By intervals are non-...
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Equivalence between monotonicity and continuity in Alexandrov topology?
One of the natural topologies on any poset $(X, \le)$ is the Alexandrov topology in which open sets are precisely upper sets: $$\tau=\{G\subseteq X \mid x\in G\land x\le y \implies y\in G\}.$$
Today I ...
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Check if this function $f(x, y) = \frac{x^2 \cdot y^3}{x^2+y^2}$ is continuous [duplicate]
I have to prove that this multivariable function is continouous,
I've used this restriction: $$(x, y) = (0, m\cdot x)$$
$$f(x, y) = \dfrac{x^2 \cdot y^3}{x^2+y^2}$$
this is valid if $(x, y) \neq 0$.
...
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is $\exists s>0 \quad \forall K\in \mathbb{R} \quad \exists \alpha > K \quad \bigg| \int_{a}^{b}f(t)\sin\alpha t dt \bigg| \ge s$ true or false [closed]
Question: We have real numbers $a,b$ and $a<b$. Given $f:\mathbb{R} \to \mathbb{R}$ is a function whose derivative is continuous everywhere.
$$\exists s>0 \quad \forall K\in \mathbb{R} \quad \...
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Determine the continuous function such that $\lim\limits_{x\rightarrow a} \frac{f(x)}{x-a}= e^{a},$ and $(x-a)(y-a)f(x+y)=(x+y-a)f(x)f(y)$
Determine the continuous function $f:\mathbb{R}\to\mathbb{R}$ with the following properties:
$$\lim\limits_{x\rightarrow a} \frac{f(x)}{x-a}= e^{a},$$ where $a$ is a real valued constant;
$$(x-a)(y-a)...
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oscillatory discontinuity
Definition of oscillatory discontinuity
A function f: [a,b] $\to\mathbb R$ is said to be discontinuous at $a\in \mathbb R$ when, $\lim_{x \to b} f(x)$ does not exists, but there exists a ...
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Completeness of the measure space assumes continuity a.e. to imply measurability of functions
I am studying Real analysis and I want to prove (or disprove) that (this is inspired by the fact that every continuous function is measurable)
Problem If $f:X\rightarrow \mathbb R$ is continuous a.e. ...
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If $S$ is a compact space and $f: S \rightarrow T$ is continuous then $f(S)$ is compact in $T$ [duplicate]
Could you check and correct my proof? Thank you !
Firstly we prove that if $V$ is an open subset of $T$ then $f^{-1}(V)$ is a open set in $S$. Indeed, take any $x\in f^{-1}(V)$ then $f(x) \in V$. ...
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Why do we have to check for the continuity of partial derivatives for differentiabilty?
According to the Differentiability theorem, we need the partial derivatives to exist and to be continuous (at a point) in order for $f$ to be differentiable at that point.
But when we perform the ...
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Check my solution for the continuity of this function
$$f(x, y) = \begin{cases} \frac{x^4y^3}{x^8 + y^4} & (x, y) \neq (0, 0) \\\\ 0 & (x, y) = (0, 0)\end{cases}$$
After having tried a couple of paths, getting $0$ as a limit, I took it as a good ...
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Weak convergence on separable and complete product space
I read a paper in which the authors seem to have a simplified definition of convergence in distribution of random variables in a product space. The paper itself is very specific, so I can link it but ...
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Continuity of partial derivatives
I'm really asking sorry for this question because I know it's a poor question, but I need a final anwer about.
Suppose I have a function $f(x, y)$, defined in some way when $(x, y) \neq (0, 0)$ and ...
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Doubts about discontinuous $\to $ existence of derivatives.
Maybe it's trivial, but I need certaintes about this. Consider
$$f(x, y) = \begin{cases} \dfrac{2xy^2}{x^2+y^2} & (x, y) \neq (0, 0) \\\\ 0 & (x, y) = (0, 0) \end{cases}$$
I showed that the ...
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Fiber of continuous surjection from higher to lower dimensional unit balls
Lets $B_k=\{x\in\mathbb{R}^k|\Vert{x}\Vert\le1\}$ denotes the k-dimension closed unit ball.
Suppose $f:B_n\rightarrow{B_m}$ is a continuous surjective map where m<n. For an interior point y of $B_m$...
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$\lim_{x \to \infty} g(x)f(x) = 0 $ for $g \in C^{1}([0,\infty))$ and $f \in L^{2}(0,\infty)$. [closed]
Let $g \in C^{1}([0,\infty))$ and $f \in L^{2}(0,\infty)$. Assume that $g$ is a positive, decreasing function. Can I assume that
$$
\lim_{x \to \infty} g(x)f(x) = 0 ??
$$
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Expanding function to continuously differentiable function using homogeneity
Let $f: \mathbb{R}^{n} \setminus \{0\} \rightarrow \mathbb{R}$ be a continuous differentiable and homgenuous function of degree $\alpha$. First we had to show with use of the definition of the partial ...
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prove that $f$ cross the x-axis at every point in the Cantor set and that f is continuous
Let $f$ be the function defined as follows.
Let C be the Cantor set and let $I_{n,k}$ be the open intervals removed in increasing order of left endpoint at the nth step. So there are $2^n$ $I_{n,k}$'...
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How can i compute this limit related with Riemann sums?
How can i compute this limit: $\lim_{n \to +\infty} \sum_{k=1}^{n} f(\frac{k-1}{n})(f(\frac{k-1}{n})-f(b_k ))$, where f:[0,1]->R, f differentiable function with continuous derivative and $\int_{\...
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Proving that a given function is continuous in a given set.
Consider the function $q(x) = \frac{1}{2} \| x \|^2$ defined in $C := \{ x \in \mathbb R^n \colon Ax = b \},$ for some matrix $A \in \mathbb R^{m \times n}$ and $b \in \mathbb R^m$ such that the ...
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Is this map $f:\prod_{k\geq 1} \{0,1,2,3\}\rightarrow \prod_{k\geq 1} \{0,1\}$ continuous?
Let me define $f:\prod_{k\geq 1} \{0,1,2,3\}\rightarrow \prod_{k\geq 1} \{0,1\}$ where I map the entries of a sequence $x\in \prod_{k\geq 1} \{0,1,2,3\}$ to the following points: $$0,1\mapsto 0,~~2,3\...
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Prove that $\phi$ surjective and open as well as closed map.
Let $\phi:\mathbf{C}^{3}\rightarrow \mathbf{C}^{3}$ be the map $(x,y,z)\mapsto (x+y+z,xy+yz+zx,xyz)$. Prove that $\phi$ surjective and open as well as closed map.
My Attempt: Surjectivity follows from ...
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Continuity of function $f(t,0) = f(0,t) = 0$ and $f(x,y) = 1$ for all other points
Assume we have a function $f: \mathbb R^2\to \mathbb R$, such that $f(t,0) = f(0,t) = 0$ for all $t\in \mathbb R$ and $f(x,y) = 1$ for $x \neq 0$ and $y\neq 0$.
I was told that the function is ...
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How to show a function is continuous in a given interval?
How to show a function is continuous in a given interval?
This might be a very basic question but I'm very new to calculus and I intend to prove that a given function is continuous in a given closed ...
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Is this theorem the same as uniform limit theorem, and why does my proof seems to be wrong, am I misunderstanding the notation?
Suppose $f_n→f$ uniformly, with $f_n$ continuous for all $n$. Then $f_n(x)\to f(x_0)$ as $x\to x_0$
This is the theorem I got from notes
My key attempt to show this question is as follows:
$$\lvert ...
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Another question on the procedure of majorization
I know I already asked a question about majorizations, but trust me when I say I really need to get this. Please don't whip me too much.
Let's say I need to prove that the function $$ f(x, y) = \begin{...
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