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 path in $\mathbb{R}^{n}$ passes through $D^{n}$ finitely many times.

I'm facing an algebraic topology exercise, and I only need to prove that in the title to finish it: Let $f: I \to \mathbb{R}^{n}$ a continuous path that passes through the unit sphere $D^{n} = \{ x ...
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132 views

Let $f$ be a function such that every point of discontinuity is a removable discontinuity. Prove that $g(x)= \lim_{y\to x}f(y)$ is continuous.

Let $f$ be a function with the property that every point of discontinuity is a removable discontinuity, i.e., $\lim_{y \to x}f(y)$ exists for all $x$, but $f$ may be discontinuous at some (even ...
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1answer
49 views

How do I go about figuring out delta-epsilon proofs?

I'm going through Bert Mendelson's Introduction to Topology on my own. In fact, I've tried to go through it several times and I always get stuck somewhere. This time it's on limits. I think I ...
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30 views

Proving that the function $\rho$ which sends a lifting of a circle map to its rotation number is continuous.

Let $\mathcal{L}$ denote all circle maps of degree one with nondecreasing liftings (a map $f \in \mathcal{L}$ is of degree one if its lifting $F$ satisfies $F(x+1)=F(x)+1$) . I need to prove that if ...
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1answer
31 views

Lipschitz's continuity general question

Prove that lipschitz continuous (in its domain) function $f(x)$ defined on a bounded set, has bounded range. I do not know whether this is not directly implied by the definition, nevertheless I do ...
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1answer
34 views

Lipschitz continuity of $\frac{1}{x}$ and $x^2$

Show that function $f(x)=\frac{1}{x}$ fulfills lipschitz continuity on all rays $(\epsilon, + \infty), \epsilon>0$ whereas does not fulfill lipschitz continuity on ray $(0, +\infty)$ Show that ...
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1answer
20 views

Show that $m(\Gamma)=0$, where $\Gamma$ is a curve $y=f(x)$

Suppose $\Gamma$ is a curve $y=f(x)$ in $\mathbb{R}^2$, where $f$ is continuous. Show that $m(\Gamma)=0$. Hint: Cover $\Gamma$ by rectangles, using the uniform continuity of $f$. If the ...
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1answer
155 views

Proving Riemann Integrability of a function with countably many discontinuities? (No measure theory)

Let $E = \{1/n : n \in \mathbb{N}\}$, and suppose $f: [0,2] \rightarrow \mathbb{R}$ is continuous everywhere but $E$. Prove that $f$ is Riemann integrable. I know the Lebesgue criterion for ...
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26 views

Confused between the isomorphism and continuity of $M_n\mathbb{R}$ with $\mathbb{R}^{n^2}$

I've managed to confuse myself while doing a few excercises in topology where I was asked to find the closure of a $M_2(\mathbb{R})$. The following are the questions I have. Is there a continuous ...
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1answer
80 views

Using epsilon - delta to prove discontinuity of f(x)= non-integer part of x

Let $f : R \rightarrow R$ be the non-integer part of $x$. Use the $ε-δ$ definition of continuity to show that $f$ is not continuous. I have an idea of what this function looks like but am struggling ...
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21 views

$f,g \in [0,1] , f<g $ , when is $U:=\{h \in C[0,1]:f(t)<h(t)<g(t), \forall t \in [0,1] \}$ a ball in $C[0,1]$ with respect to the sup metric

Let $f,g:[0,1] \to \mathbb R$ be continuous functions such that $f(t)<g(t),\forall t \in [0,1]$ , then under what additional conditions on $f,g$ can we conclude that $U:=\{h \in ...
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0answers
26 views

Continuity of partial derivatives and continuity of the function

Let $f: \mathbb{R^2} \to \mathbb{R}$ a function which partial derivatives exist near a point P. Suppose one of the partial derivatives is continous in P. Is this enough for the function to be ...
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1answer
25 views

Continuity of operators

Let $(T_t)_{t \ge0}$ be a family of operators(not necessarily bounded, but all defined on the same domain) and now we have the property $$t \rightarrow 0^+ \Rightarrow ||T_t^2 -T_0^2|| \rightarrow ...
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2answers
77 views

Continuous functions in a metric space using the discrete metric

Let X be any set and define $d: X \times X \to \Bbb R$ by $d(x,y)= \begin{cases} 0 & x=y \\ 1 & x \neq y \end{cases}$. Classify all continuous functions $f: X \to X$ using the discrete metric ...
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1answer
29 views

How to show that $f$ is not totally differentiable, using formal definition

Given that $$f(x,y)= \begin{cases} \frac{xy}{x^2+y^2}, & \text{if $(x,y)\neq0$}.\\ 0, & \text{if $(x,y)=0$} \end{cases}$$ I want to show that $f$ cannot be totally ...
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1answer
51 views

Given $f(x)=\big(\frac{1}{x}\big)^{1/3}$. Is the area bounded by the function and the $x$ axis finite?

Consider the function $f(x)=\Big(\frac{1}{x}\Big)^{1/3}$ with $x\in[-1,1]$. I want to find out wether the area bounded by the function and $x$ axis is finite? Using simple strategy (i.e integrating ...
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2answers
39 views

Where does the function $f(x) =\Big[\frac{1}{2}*x\Big]$ contain discontinuities, left or right continuous?

I'm trying to get a grasp on this problem here. I'm going through a calculus textbook to prep myself for a tutoring job. However, i came across this one problem i couldn't seem to make sense of. ...
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3answers
75 views

Continuity of a piecewise function of two variable

I'm given this equation: $$ u(x,y) = \begin{cases} \dfrac{(x^3 - 3xy^2)}{(x^2 + y^2)}\quad& \text{if}\quad (x,y)\neq(0,0)\\ 0\quad& \text{if} \quad (x,y)=(0,0). \end{cases} $$ It seems ...
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79 views

$\log$ is continuous

I'm trying to prove that the complex logarithm function is continuous using this theorem, but I'm hitting a snag in part of the proof. Let $D=\Bbb C\setminus(-\infty,0]$. The claim is that the ...
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1answer
27 views

Prove that there is a point $z \in [a,b]$ such that $f(z)= {f(x_1)+ \dots + f(x_n)\over n} $

Let the function $ f : [a,b] \to \mathbb{R} $ be continous, such that $ f(a) \neq f(b)$. For $ n \in \mathbb{N}$ let $ x_1,x_2,...,x_n $ be points in $[a,b]$. Prove that there is a point $z \in ...
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1answer
30 views

Consider the following statement regarding uniform continuity of a function

Given $A \subseteq \mathbb R^n$, if $G \colon A \to \mathbb R^m$ is a uniformly continuous function, then given $\epsilon$ there is $\delta$ so that if $B\subseteq A$ and ...
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1answer
82 views

Real Analysis Proof Continuity - lim sup anbn≤(lim sup an)(lim sup bn) and …

1) If an ≥ 0 and bn ≥ 0, prove that lim sup anbn ≤ (lim sup an)(lim sup bn) 2) If {an} and{bn} are non-negative sequences and {bn} converges, prove that lim sup anbn = (lim sup an)(lim bn). I am not ...
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566 views

Does there exist a continuous and differentiable function which isn't smooth?

As I understand, a smooth function is continuously differentiable. But if I have a function which is continuous AND differentiable, I cannot automatically say that it is smooth. For it has to be so ...
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1answer
41 views

Show f is strictly increasing on [a,b] if cts on [a,b] and differentiable on (a,c) and (c,b)

Where a < c < b and f'(x) > 0 for all x in (a,c), (c,b). Using the Mean Value Theorem I have shown that the f is strictly increasing on [a,c) and (c,b]; my question is: how do I involve c in ...
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0answers
26 views

Piecewise absolutely continuous functions: norm implication

I have the following doubt: Consider a sequence of functions $(f_k)$, $k=1,2,\ldots$ which are piecewise absolutely continuous functions $f_k: [a,b] \rightarrow \mathcal{R}^n$ with a finite number of ...
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1answer
25 views

Understanding a certain limit

Suppose we've got a function $f(x,y):=\frac{y^2+2x}{y^2-x}$ defined on the real plane expect for the set $y^2=x$. Now we want to see whether $f$ can be made continuous in $(0,0)$. My work: I take ...
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1answer
40 views

Bilinear mapping

The question is as follows, and I have added my attempt at the proof, but I don't have much information about bilinear mappings. Rudin only provided a definition. I was hoping that you all might be ...
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1answer
32 views

Compactness of the set of points where a continuous function achieves a local maximum

Let $(K,d)$ be a compact metric space, and $f:K\rightarrow \mathbb{R}$ be a continuous function on $K$. Define: $$M=\left \{ x\in K :\text{$f$ achieves a local maximum in $x$} \right \}$$ I need to ...
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1answer
42 views

function not continuous, partial derivatives exists -> partial derivatives not continuous

I'm a bit confused about this. I know that if all partial derivatives exist it's not necessary for function to be differentiable. Usual examples for non differentiable function for which all partial ...
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34 views

How to show that $f_n(x_n) \to f(x)$?

Define $f_n(x) = \int_0^x |T_n\left((|s|-\frac 1n)^+ + \frac 1n\right)|^{-\frac 12}\;\mathrm{d}s$. Here $T_n(x) := x$ if $|x| \leq n$ and $T_n(x) := n$ otherwise. We have that $f_n(x) \to f(x):= ...
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2answers
40 views

differentiability at a point (0,0) based on partial derivatives

For $$ f(x,y)=\begin{cases} y^2 sin\left(\frac{x}{y}\right) & \text{if } y\neq0 \\ 0 & \text{if } y=0 \end{cases}$$ i've shown that it is continuous and that the partial derivatives ...
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1answer
22 views

Extending this function to $\mathbf R$

I have the following function that is defined for $x$: zero or positive (natural number) power of 10. $$\sum_{k=0}^{\log_{10}x}\left({1\over 2}\right)^{k+1}$$ $$1\to 1/2$$ $$10\to 3/4$$ $$100\to ...
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1answer
44 views

Coordinate-wise continuity

I'm a bit confused by cooridnate-wise continuity. What would it mean for the function: $f(x,y)= \begin{cases} \frac{xy}{x^2+y^2} & (x,y)\neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$ to be ...
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0answers
19 views

Continuity of inverse function at endpoints

Let $f$ be a strictly increasing continuous function on a closed interval $[a, b]$, let $c = f(a), d = f(b)$, and let $g:[c, d] → [a, b]$ be its inverse. Then $g$ is a strictly increasing continuous ...
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17 views

Continuous composed with differentiable

If $f(x)$ is $C^\infty$ and $g(x)$ is bounded and continuous does that imply that $f(g(x))$ is differentiable
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1answer
35 views

Continuity of $f(x,y)=x^3\sin\left(\frac 1x\right)+y^2$ at $(0,0)$

Consider $$f(x,y)= \begin{cases} x^3\sin(1/x) + y^2\quad & x\ne 0 \\ y^2 &\text{otherwise}\end{cases}$$ Prove that $f$ is continuous at $(0,0)$. I do not have an experience in proving ...
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2answers
143 views

The difference set of two continuous functions is open

Let $f,g:X\to Y$ be continuous functions on topological spaces $X,Y$. Then the set $\{f\ne g\}:=\{x\in X:f(x)\ne g(x)\}$ is open. This statement seems to be true under some mild extra hypotheses ...
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1answer
99 views

Example of non-differentiable continuous function with all partial derivatives well defined

Give an example of a function $f : \mathbb{R}^3 \to \mathbb{R}$ such that the partial derivatives exist at $(0,0,0)$, and $f$ is continuous at $(0,0,0)$, but it is not differentiable at $(0,0,0)$. Any ...
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32 views

differentiability and continuity in R3

Prove that if a function is differentiable at $(a,b,c)$ in $\mathbb R^3$ then it is continuous at $(a,b,c)$. I tried to imitate the proof that if $f$ is differentiable at a specific point in $\mathbb ...
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1answer
17 views

If $u_n \to u$ in $L^2(0,T;L^2(\Omega))$ and $f_n \to f$ uniformly, does $f_n(u_n) \to f(u)$ in $L^2(0,T;L^2(\Omega))$?

Let $\Omega$ be an unbounded domain. Suppose we have $u_n \to u$ in $L^2(0,T;L^2(\Omega))$. Let $f_n\colon \mathbb{R} \to \mathbb{R}$ be a sequence such that $f_n \to f$ uniformly. We know that ...
3
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2answers
29 views

$\phi(x)-\psi(x)=(\phi(x_0)-\psi(x_0))e^{-\int_{x_0}^x a(t) dt}$

I am looking at the following exercise: If $\phi, \psi$ solutions of the differential equation $y'+a(x)y=b(x)$ on an interval $I$, where $a,b$ continuous on $I$ and $x_0 \in I$, show that: ...
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99 views

Proving the function $ f $ is continuous on $ [0,1] $

I'm trying to prove that the following function $ f $ is continuous on $ [0,1] $. The function $ f:[0,1]\rightarrow [0,1] $ is defined as follows. Let $ x\in [0,1] $. Then $ x= \sum\limits_{n = ...
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1answer
32 views

Lower semicontinuity of ${\dot{H}}^1$ norm

I have a in $H^1(\mathbb{R^N})$ uniformly bounded sequence $u_n \in H^1$. I also know $u_n\to u$ in $L^p$ for every $2\leq p < 2^\ast$, where $\ast$ means the Sobolev exponent. Can I conclude that ...
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1answer
29 views

Decomposition of a Hölder continuous function

Suppose $f:\mathbb{R}^3\rightarrow\mathbb{R}$ is a bounded, Hölder continuous function. Let $0<r<s$. How is it possible to write $f=f_1+f_2$ with $f_1$ Hölder continuous and vanishing outside ...
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1answer
34 views

Is this sequence of functions (involving antiderivative and truncations) uniformly convergent?

Define for $m \in (0,1)$ fixed the sequence $$f_n(x) := \int_0^x |T_n\left((|y|-\frac{1}{n})^+ + \frac{1}{n}\right)\text{sign}(y)|^{m-1}$$ where we define $T_n(y) =y$ if $|y| \leq n$ and otherwise ...
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2answers
83 views

Finding $\lim_{(x,y)\rightarrow (0,0)} \frac{\tan(x^2+y^2)}{\arctan(\frac{1}{x^2+y^2})} $

I'm having trouble understanding how the $\displaystyle\lim_{(x,y)\rightarrow (0,0)} \frac{\tan(x^2+y^2)}{\arctan(\frac{1}{x^2+y^2})}$. I used the product law to set it up as ...
0
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1answer
39 views

Continuity of $f:\mathbb R^2 \rightarrow \mathbb R$

Consider $f:\mathbb R^2 \rightarrow \mathbb R$ $f(x,y)=\begin{cases} xy,\text{ if } xy > 0\\ 0, \text{ if } xy \le 0 \end{cases} $ at which points of $\mathbb R^2$ is $f$ continuous? My ...
0
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1answer
22 views

for which constants function is continuous

$$f(x)=(x^2+a^2)^{(1/2)}\;\;|x|>1$$ $$f(x)=ax^2+bx+c\;\;|x|\le 1$$ Function may not be continuous in $1$. So I have to check for which $a,b,c$ ...
0
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1answer
63 views

Suppose that $ f:I\rightarrow R $ is uniformly continuous--prove that the righthand limit at the endpoint exists

Suppose $ f $ is a uniformly continuous function mapping from an open real interval $(a,b)$ into the real numbers. Then the limit as $x $ approaches $ a $ from the right of $ f(x) $ exists. I'm ...
3
votes
2answers
56 views

$f:[0,\infty) \rightarrow \mathbb{R}$ is continuous, prove uniform continuity on $[0,\infty)$ given…

$f:[0,\infty) \rightarrow \mathbb{R}$ is continuous. Given $f$ satisfies $\epsilon>0$ there exists $M_{\epsilon} > 0$ such that $x>M_{\epsilon} \implies |f(x)| < \epsilon$ Prove uniform ...