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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Non-decreasing functions and continuity

I have the following situation: $f\colon\mathbb{R}\to\mathbb{R}$ is a non-decreasing $g\colon\mathbb{R}\to\mathbb{R}$ is defined as $\ g(x):=\lim_{t\to x^+}f(t)$ I have proved that also $g$ is ...
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Microcontinuous vs Continuous

I've been studying infinitesimals and came upon the idea of uniformly microcontinuous functions. My question is: if a function $f^*: \mathbb{R}^* \to \mathbb{R}^*$ the natural extension of $f: ...
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12 views

Upper Hemi Continuity

I am trying to understand the upper hemi continuity property and, by its wikipedia definition, it seems to me that a correspondence that is everywhere a continuous functions, except at one point, ...
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46 views

Differentiability conditions for a piecewise function

So this is an analysis class, and we just started the unit on differentiability -- however I missed the class. Can someone start me off with a good real analysis definition for differentiability of ...
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27 views

Continuity of an application between function spaces.

I'm trying to prove the following statement... Let $f:[a,b] \times \mathbb{R} \to \mathbb{R}$ a bounded and continuous function, $t_{0} \in [a,b]$, $x_{0} \in \mathbb{R}$, $r>0$ and $$B= \{ x ...
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What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
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19 views

Discontinuous functions with finite Fourier series approximation?

Yesterday I posted a question regarding the computation of complex Fourier coefficients for the functions $$f(t) = \sin(2 \pi t)$$ $$f(t) = |\sin(2 \pi t)|$$ where $0 \leq t \leq 1$. The first ...
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1answer
48 views

Where is piecewise dirichlet function with $|x|^2$ continuous or differentiable?

If $|x|^2$ is continuous and differentiable on all of $\mathbb{R}^n$ (already shown differentiability by showing all $n$ of its partial derivatives are continuous), then... Question: For the function ...
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bounded monotone continuous function is uniform continuous [duplicate]

If $f: \mathbb R \to \mathbb R$ is bounded, increasing and continuous. Does $f$ have to uniform continuous? I know the answer is yes if $f$ has domain to be any open interval, say $(0,1)$. But I ...
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37 views

Prove that $f^y$ and $f_x$ are Lebesgue-integrable

Let $f:\Bbb R^2\to \Bbb R$ given by: $$f(x,y) = \begin{cases} \frac{x^2-y^2}{(x^2+y^2)^2} & \text{if $(x,y)\in(0,1)\times(0,1)$} \\ 0 & \text{if $(x,y)\not\in(0,1)\times(0,1)$} \\ ...
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36 views

Continuous function existence

Prove that there is at most one continuous function on $[0,2]$ that satisfies: $$v(x)=f(x)+\int_0^2 e^{-(x-y)^2} \cos(0.3v(y)) \, dy$$ I don't know how to estimate this integral...
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49 views

Spivak problem on property of continuous functions.

Ok so problem goes like this: If f is continuous on [0,1] and f(x) is in [0,1] for each x.Prove that f(x)=x for some x. My proof goes like this but I am not quite sure of my result. Let ...
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46 views

Continuous images of open sets

In trying to prove that the graph of a continuous map of compact Hausdorff spaces, $f:X\to Y$ is compact, I stumbled on this problem: Let $f:X\to Y$ be a continuous function, $U$ and $V$ open in $X$ ...
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Convolution is continuous map

I can prove this when $f$ is assumed as continuous function but without assuming continuity i got confused. Suppose $ p \in (1, \infty) $ and $q$ is its conjugate exponent. Prove that if $f\in ...
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What is the difference between “differentiable” and “continuous”

I have always treated them as the same thing. But recently, some people have told me that the two terms are different. So now I am wondering, What is the difference between "differentiable" and ...
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66 views

Show a function is not continuous at a point

$$ f(x,y) = \begin{cases} \dfrac{x^2 y^4}{x^4 + 6y^8}, & \text{if }(x,y)\neq(0,0) \\ 0, & \text{if }(x,y)=(0,0) \end{cases} $$ For the definition of differentiability, I have: $$\lim_{h ...
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36 views

Totally differentiable function - definition

I know for a function of several variables, if all partial derivatives exist and they are continuous at and around a point $a$ then the function is totally differentiable at that point. I ...
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32 views

Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)

Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$) i) Suppose $\sum a_i$ converges. Must $f'(0)$ exist? ii) Suppose $f(0) = f'(0) = 0$. ...
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95 views

Continuity proof for exponential

Show that $f(x) = e^x$ is continuous using the epsilon-delta definition. I can't seem to write down anything meaningful...
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77 views

Discontinuity of the characteristic function

Let $A \subseteq \mathbb{R}^n$. Let $f(x) = \chi_A $ be the characteristic function, and put $D = \{ x : f(x) \; \; \text{is discontinuous} \} $. Then $\partial A = D $. MY try: Let $y \in D $. ...
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18 views

Continuity on square set

Prove $f(x)=e^{-x-y^2}$ is continuous on Q. How can I show the continuity of a multivariable function from the definition?
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29 views

Proving a if a property holds for a dense set then it holds on the field that the set is a subset of.

I am currently studying for my analysis exam and have come across this question, I can't seem to grasp the idea of a "dense set" especially with the definition given in the question. When I read it, ...
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171 views

What does Structure-Preserving mean?

A very basic definition in category theory is the definition of morphism between objects. If the category is a construct, i.e., a category $\mathcal C$ equipped with a faithful functor $U\colon ...
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A problem of weak* continuity in relation with semigroups

Let $(\Omega,\Sigma,\mu)$ be a probability space. Let $\mathcal{A}$ ba a $\sigma$-subalgebra of $\Sigma$. We denote by $\mathbb{E} \colon L^\infty(\Sigma) \to L^\infty(\mathcal{A})$ the associated ...
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Continuity of a parametric integral (where the integrated function is discontinuous)

For all $t\in\mathbb{R}$ consider $$F(t):=\int_\mathbb{R}e^{-x^2/2}\log|t+e^x|\,dx \;.$$ I managed to show that $F(t)$ is well-defined and finite for every $t$. I would like to show that $F$ is ...
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36 views

Continuity of integral of continuous functions

Let $f\in L^1(\mathbb{R})$. Show that the function $g$ defined on $\mathbb{R}$ by $$ g(x) = \int_{\mathbb{R}} \sin(xy)f(y)dy$$ is well defined and continuous on the real line. So I want to prove ...
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48 views

If the sum of two functions is continuous at a point, can one function be continuous and the other not

I got this question: Let $f$ and $g$ be functions such that $f+g$ is continuous at $a$, Must it be the case that both $f$ and $g$ are continuous at $a$ or that both $f$ and $g$ are discontinuous at ...
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90 views

Continuous function $f: \mathbb R \to \mathbb R $ such that the set { $ x \in \mathbb R : f(x)<0$ } is singleton

I am in desperate need of an example of a continuous function (if exists) $f: \mathbb R \to \mathbb R $ such that $ f(x) <0 $ for exactly one $x \in \mathbb R $ ; please help .
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multivariable calculus question norm

Given vector space C([a,b],$ \mathbb{R} $) of continuous functions of [a,b] in $ \mathbb{R}. $ Prove that the function $ \left \| f \right \|_{1}=\int_{a}^{b}\left | f(t) \right |dt $ is a norm. Also ...
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Show that a map is a continuous bilinear form on $H^1(0,1)$ space

Let $u,v \in H^1(0,1) = \{f : (0,1) \longrightarrow \mathbb{R}, f,f' \in L^2(0,1) \}$, show that $$a(u,v) = \int_0^1 (u'v' + uv)\; dx$$ is a continuous bilinear form.
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modern analysis: integrals and continuity

Let $$f(x) = \sum_1 ^\infty n*e^{-nx}$$ Where is $f$ continuous? Compute $$\int_1^2f(x) dx$$ I am having trouble proving where $f$ is continuous. For the second part, so far I have been able to ...
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Prove that $e^{x}-2\cos(x) = 0$ where $x\in(0,1)$ has solution.

Prove that $e^{x}-2\cos(x) = 0$ where $x\in(0,1)$ has solution for $x$. I'd like to do this without derivatives, just using limit definition and function continuity. To begin, we could rewrite this ...
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1answer
16 views

Convergence and continuity

${f_n}$ is continuous on $\mathbb{R}$, and $f_n \to f$ uniformly on every interval $[a,b]$. Prove $f$ is continuous on $\mathbb{R}$. I know that it must be the case that $f$ is continuous on $[a,b]$. ...
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Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
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1answer
74 views

$\exists f:\mathbb{R}\rightarrow \mathbb{R},$ continuous, non-constant, with uncountably many extrema?

I couldnt think of any; by intuition I don't think any can exist, but I can't figure out how to prove it. If it existed then the set of extrema would have to be uncountable but I think this might ...
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1answer
64 views

Proof that $f(x)=x^{1/n}$ is continuous.

Here's what I've done: According to the definition, a function is continuous at $c$ if, for any $\epsilon>0$, there exists a $\delta>0$ so that, if $|x-c| < \delta$, then $|f(x)-f(c)| < ...
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Convergence of a sequence pointwise a.e. on a manifold given that it converges on a reference manifold

Let $\Gamma_t$ be a compact hypersurface for each $t \in [0,T]$. Let $Q=\Gamma_0\times(0,T)$ and $$Q_T :=\bigcup_{t \in (0,T)} \Gamma_t \times \{t\}.$$ For each $t \in [0,T]$, suppose that ...
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1answer
34 views

Prove $f$ not continuous at SEEMOUS Contest

Let $n$ be a nonzero natural number and $f:\mathbb{R}\to\mathbb{R}\setminus\{0\}$ be a function such that $f(2014) = 1 − f(2013)$. Let $x_1,x_2,x_3,...,x_n$ be real numbers not equal to each other. ...
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1answer
30 views

Prove $g$ is continous on metric space.

For:$(X,d)$ is a metric space , $f:X\to X$ is a continous function and $g:X\to R, x\mapsto g(x)=d(f(x),x)$. Prove that $g$ is a continous function. Definition: $f$ is continous at $x_0$ if only if ...
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39 views

Seemingly easy analysis problem but unsure how to proceed.

if $f(x)=\frac{1}{x+2}$ then $f(x)=1-(x+1)+(x+1)^2+T$ for some $x_0$ between $x$ and $-1$ where $T=-\frac{(x+1)^3}{(2+x_0)^4}$ I'm not sure how to proceed in solving this problem. We recently ...
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96 views

How do I prove that, for this given function, $f$ is continuous at $a$ iff $a=-1$?

I'm given the function: $f(x)=\left\{ \begin{array}{lr} -x & : x \in \mathbb{Q}\\ x+2 & : x \notin \mathbb{Q} \end{array} \right.$ How would I (at least go about ...
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1answer
26 views

Construct two functions based on big O constraint

I'm doing an algorithm problem goes like this. Construct two functions $f$, $g$ : $\mathbb{R}^+\rightarrow\mathbb{R}^+$ satisfying, $f$, $g$ are continuous; $f$, $g$ are monotonically increasing; ...
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66 views

Proving a function is continuous using preimages

I want to prove that f is continuous using the preimages of open subsets here. Never worked with pre images before -- can anyone help? (also would love a good definition of a preimage).
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1answer
82 views

Prove the continuity in the composition function.

If $g$ is continuous at $c$ and $f$ is continuous at $g(c)$ Then prove that $(f\circ g)$ is continuous at c. To prove this I have done something: Given: $$\lim_{x\to c}g(x)=g(c) \tag 1 $$ ...
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47 views

Is function continuous $x\sin(y)/(x^2+y^2)$

I have the following function and I can't seem to prove that it is not continuous: $ f(x,y) = \begin{cases} 0, & {(x,y) = (0,0)} \\ x\sin(y)/(x^2+y^2), & \text{else} \\ \end{cases}$
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55 views

Bounded Jacobian implies uniform continuity

I am trying to solve the following problems but I am not sure what the difference between the 2 problems is. 1) Prove that is $U = B_r(x)$ (open ball centered at $x$ with radius $r>0$) is an open ...
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1answer
58 views

$(6+6 \cos x) / \sin x$ is continuous on what interval and why?

$f(x)=\frac{6+6\cos x}{\sin x}$ is continuous on the interval $(n\pi,(n+1)\pi)$ where $n$ is an integer. I understand the continuous interval concept, but I don't understand why that specific ...
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40 views

continuity and differentiability of function of two variables

Let $f(x,y)$ be $$f(x,y): \begin{cases} x & \text{for } y = 0\\ x-y^3\sin\left(\frac{1}{y}\right)& \text{for } y \neq 0\end{cases} $$ then check continuity and differentiability at $(0,0)$. ...
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2answers
42 views

What kind of functions can be Riemann integrable?

I have learned that every continuous, or piecewise continuous function can be Riemann integrated. But then, are there uncontinuous functions that are Riemann integrable? And if there is, can I still ...
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1answer
32 views

Let C be a circle. Show that the only subset of C homeomorphic to a circle is C itself

I am trying to answer the question stated in the title. The hint in my book says to realize that for any z on the circle C{z} is still connected. I believe I can deal with case that shows that C{z} ...