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) ...

learn more… | top users | synonyms (1)

3
votes
1answer
144 views

If $f$ takes Cauchy sequence to Cauchy sequence then $f$ is continuous [duplicate]

If $f:X\to Y$ takes Cauchy sequence to Cauchy sequence then prove that $f$ is a continuous function. Let $x_n$ be a sequence in $X$ such that $x_n\to x\implies x_n$ is Cauchy $\implies f(x_n)$ is ...
-1
votes
1answer
28 views

If $f$ is a map from a topological space $Y$ to a metric space $X$, to prove that $f$ i

If $f$ is a map from a topological space $Y$ to a metric space $X$, to prove that $f$ is continuous at y, is it enough to show that for all $\epsilon >0$, there exists $V_{y}$ (neigbhourhood of $y$ ...
1
vote
0answers
22 views

Check uniform equicontinuity of a function family

I am struggling to prove or disprove that the following function family is uniformly equicontinuous. $$F = \{f \in C^1[0,1]: \forall x \text{ } |f(x)| + \sqrt x |f'(x)| \leq 1 \}$$ First I tried to ...
7
votes
2answers
694 views

Can we always multiply some function that is not differentiable everywhere with function that is to obtain differentiable product?

First of all, I think that before stating the general question it would be okay to make some concrete example of what do I have in mind. Let us take the function $f(x)=|x|$. We could write this ...
4
votes
1answer
66 views

Let $f$ be a continuous and open map from $\mathbb R $ to $\mathbb R$.Prove that $f$ is monotonic.

Let $f$ be a continuous and open map from $\mathbb R $ to $\mathbb R$.Prove that $f$ is monotonic. Suppose that $f$ is not monotonic.Then $\exists a,b;a>b$ such that $f(a)<f(b)$ and $c,d;c<...
0
votes
1answer
22 views

Problem in Topology related to continuous functions

Let $\mathbb{R-Q}$ be the subspace of $\mathbb{R}$ with the usual metric. Is there a function, $f:\mathbb{R-Q} \rightarrow \mathbb{R-Q}$ such that f is continuous and $f$ does not have a fixed ...
2
votes
2answers
59 views

Prove that if $f: A→B$ and $g: B→C$ are continuous, then so is $g\circ f: A→C$.

I wanted to start by using the definition of continuity. But my definition is only for $f: A→R$, so I'm unsure as how to write $\lim_{x\to a}f(x) = f(a)$ for $f: A→B$ and $g: B→C$. Would it be ...
0
votes
0answers
44 views

Uniformly Continuous Sequences

How can I show: Let {$f_n$} be a sequence of functions that are uniformly continuous on $(0,1)$. Show if {$f_n$} converges uniformly to $f(x)$ on $(0,1)$ then $f(x)$ is uniformly continuous on $(...
4
votes
4answers
542 views

Can the function defined in this way be everywhere discontinuous?

Suppose that we have some real function of a real variable $f$ defined on the set $[a,b]$ which has the properties that: 1) $f$ takes values in the set on which it is defined 2) for every $y \in [a,...
0
votes
1answer
16 views

Estimating the measure of set and measure of shifted set

Could you please help me solve the following problem: Is it true that for any $\varepsilon>0$ there exist a $\delta=\delta(\varepsilon)>0$ (depends on $\varepsilon$) such that $$ \int_{A}\frac{...
0
votes
1answer
76 views

Can the limit of piecewise linear continuous functions be some differentiable function other than linear or constant functions?

Oh, well, the title actually describes what kind of question will this question be, but let us do some warm-up before stating the question as clearly as possible. Suppose first that everything we do ...
0
votes
2answers
57 views

Is the following theorem true?

So, I am working on some analysis homework and created a lemma to help me prove something. The problem is I don't know if it's true or false, and I don't want to waste a whole bunch of time attempting ...
0
votes
1answer
37 views

Showing a function is continuous

I am trying to show that $$h\colon X \to X \times Y, x \mapsto x \times y_0$$ where $y_0$ is fixed is continuous. I have taken $U \times V \subset X \times Y$ and said that $$h^{-1}(U \times V) = \{...
0
votes
1answer
23 views

Verifying that a function is a cumulative distribution function

If I have a function $$ F(x) = \left\{\begin{array}{} 0, & \text{if } x \leq -1\\ \frac{1}{2} - \frac{x^2}{2}, & \text{if } -1 \leq x \leq 0\\ ...
0
votes
2answers
57 views

Prove that $\lim_{x\to a}f(x) = L$ if and only if $\lim_{x\to a^-}f(x) = L$ and $\lim_{x\to a^+}f(x) = L$

First, I mentioned that $\lim_{x\to a^-}f(x) = L$ if there exists a $\delta > 0$ such that $a-x < \delta$ where $|f(a)-f(x)| < \epsilon$. And that $\lim_{x\to a^+}f(x) = L$ if there exists ...
0
votes
0answers
37 views

Can you solve this Calculus temperature decay problem directly without using Calculus?

The problem is: (The rate of decrease of temperature of an object is continuous and proportional to the difference between the temperature of the object and that of the surrounding medium.) Suppose ...
0
votes
0answers
48 views

Proving inverse function theorem

I'm asking for an help to understand the proof of the inverse function theorem, in particular one part of it represented by Lemma 4.2 in van der vaart p.36 (you can find it here https://books.google....
-2
votes
1answer
74 views

Prove that $\lfloor x \rfloor \sin (\pi x)$ is continuous on $\Bbb R$ [closed]

I try to prove that $f(x) = \lfloor x \rfloor \sin (\pi x)$ is continuous on $\Bbb R$. Someone help me, please. I need it tomorrow. Thank you.
2
votes
1answer
52 views

Sequence of polynomials $p_n$ converging to a non-polynomial. Show $\text{deg}\, p_n\to\infty$

If $f\in C[a,b]$ is not a polynomial, then show that for any sequence of polynomials $p_n$ that converges to $f$ uniformly, one must have that $\text{degree of } p_n \to\infty$.
0
votes
1answer
51 views

Proving that $f'$ is measurable on $\mathbb R$ if$f$ is differentiable on $\mathbb R$

Since $f$ is differentiable on $\mathbb R$ it is then continuous on $\mathbb R$,making $f$ measurable. (this we proved in class-that continuous functions are measurable). I tried to use this to prove ...
2
votes
1answer
44 views

If f is continuous, prove $h(x)=d(x,f(x))$ is continuous

I'm trying to prove that if $f:X\to X$ is continuous, then, $h:X\to\mathbb{R}$ defined by $h(x)=d(x,f(x))$ is continuous. I want to show there exists $r>0,$ such that if $\forall p\in X,$ d$(p,x)&...
2
votes
0answers
55 views

Showing continuity of a multivariable function continuous in each variable, with $x\mapsto f(x,y)$ for fixed $y$ being equicontinuous

The problem statement is as follows. We are given a function $f:[0,1]\times [0,1] \to \mathbb{R}$. For all fixed $x$, $y\mapsto f(x,y)$ is continuous, and likewise, for all fixed $y$, $x\mapsto f(x,y)$...
1
vote
3answers
67 views

Is $f_n$ guaranteed to have a pointwise limit?

Let $f_n$ be a sequence of non-negative continuous functions on $[0,1]$ such that $\lim_{n\to \infty}\int _0^ 1 f_n(x) dx=0$. Is $f_n$ guaranteed to have a pointwise limit? I think the answer is ...
0
votes
2answers
33 views

Proving a definite integral is positive

Okay so I can't make heads or tails of this supposed solution given by my lecturer. The result we have to prove is obvious; clearly a function that is positive over an interval has a positive definite ...
0
votes
1answer
34 views

Uniformly Continuous & Dense Subsets

Suppose that $(\mathcal X,d)$ and $(\mathcal Y, \rho)$ are two metric spaces. Suppose that $(\mathcal Y, \rho)$ is complete and that $\mathcal D$ is a dense subset of $\mathcal X$. Show that if $f:...
3
votes
0answers
90 views

Convergence for a improper integral $\int^b_a fg$

Let $f$ be continuous on [a,b) such that $\int^b_a f$ converges. If $g'$ is locally integrable and has a constant sign on [a,b), prove that $\int^b_a fg$ converges. Edit: We can assume that the limit ...
0
votes
2answers
30 views

Show that P[a,b] is a strict subset of C[a,b]; in other words, there are necessarily nonpolynomial elements in C[a, b].

Let $P([a, b])$ denote the space of all polynomials on $[a, b]$. Clearly $P([a, b]) \subseteq C([a, b])$. Show that $P([a,b])$ is a strict subset of $C([a,b])$; in other words, there are necessarily ...
5
votes
1answer
106 views

Unexpectedly uniformly continuous functions

The other day in a exam, I was given the following exercise: Given $f : [0,1] \to \mathbb{R}$ continuous and such that $f(0) = 0, f(1) = 1$, let $g : \mathbb{R} \to \mathbb{R}$ be $g(x) = [x] + f(...
0
votes
1answer
48 views

Image of an open set in complex analysis

Is it true that the image $f(D)$ of any open set $D$ $\subset$ $X$ is open? Here $f$ is a continuous function, and $X$ is a topological space. Can someone explain this to me?
0
votes
1answer
65 views

If a complex function $f(z)$ is discontinuous at $z_0$ then does that imply that the derivative $f'(z_0)$ does not exist?

Moreover if the implication is correct can then one use this result to test whether a line in the complex plane is a branch cut? Say a function $g(z)$ has a branch cut along $(-\infty, 0)$. One ...
1
vote
1answer
64 views

Image of a compact set under a linear transformation

Let $f:\mathbb{R}^p \rightarrow \mathbb{R}$ be a linear transformation i.e $f(x) = a\cdot x$ with $a=(a_1,...,a_p) \in \mathbb{R}^p$. Prove: if $K=\{ x \in R^p : \|x\| \leq 1\}$ then $f(K) =[-M,M]$ ...
1
vote
1answer
59 views

Show that $f$ differentiable implies $f$ continous [duplicate]

I have to show that, if a function $f:\mathbb R\to\mathbb R$ is differentiable, it is also continous. $$\lim_{h\to0}\frac{f_{(x+h)}-f_x}{h}=f'_x\space\space\space\space\forall x\in \mathbb R$$ To ...
1
vote
1answer
73 views

In a Completely regular $T_1$ space, two disjoint sets, one compact, the other closed, can be separated by a continuous function?

Let $X$ be a completely regular $T_1$ space and let $A,B$ be disjoint closed subsets of $X$, where $A$ is compact also. Then is it true that there exist a continuous function $f\colon X \to [0,1]$ ...
8
votes
2answers
474 views

Is a continuous function locally uniformly continuous?

Assume a function, $f : X \to Y$, mapping between two metric spaces, $X,Y$, is pointwise continuous, i.e. for every $\varepsilon >0$ and $x \in X$ there exists a $\delta>0$ such that $$ \|x-x'\|...
6
votes
1answer
74 views

If a separately continuous function $f : [0,1]^2 \to \mathbb{R}$ vanishes on a dense set, must it vanish on the whole set?

Assume $f(x,y)$ is defined on $D=[0,1]\times[0,1]$, and $f(x,y)$ is continuous of each separate variables(i.e. if we fix $y$ to $y_0$, then $f(x,y_0)$ is continuous and vice versa). If $f(x,y)$ ...
0
votes
1answer
63 views

Fixed Points of Polynomial (Application of Mean Value and Intermediate Value Theorems)

The question is: A number $a$ is called a fixed point of a function $f$ if $f(a)=a$. Consider the function $f(x)=x^{87}+4x+2, x\in\Bbb R.$ (a) Use the Mean Value Theorem to show that $f(x)$ ...
2
votes
1answer
33 views

When does $\log(\lim_{x\to c} f(x)) = \lim_{x\to c} \log(f(x))$?

When does $$\log(\lim_{x\to c} f(x)) = \lim_{x\to c} \log(f(x))$$ I have seen different things from different sources. For example, this and this. Does $f(x)$ have to be continuous at c, does $\log (...
2
votes
0answers
82 views

Edelstein Fixed Point Theorem

Let $(M,d)$ be a metric space, $M$ compact. If $f:M \to M$ is continuous and weakly contractive (i.e. $d(f(x), f(y)) < d(x,y) , \forall x,y \in M$), then $\exists x_0 \in M $ unique s.t $f(x_0)=...
0
votes
1answer
35 views

Show $f: [0,1] ^{\mathbb{N}}\to [1,e]^{\mathbb{N}}: (x_{i})_{i \in \mathbb{N}} \mapsto (e^{x_{i}})_{i \in \mathbb{N}}$ is a homeomorphism

We consider the function $f: [0,1] ^{\mathbb{N}}\to [1,e]^{\mathbb{N}}: (x_{i})_{i \in \mathbb{N}} \mapsto (e^{x_{i}})_{i \in \mathbb{N}}$, where $[0,1]$ and $[1,e]$ are subspaces of $\mathbb{R}$ with ...
1
vote
2answers
30 views

Intuition: Why is continuous decay expressed as the inverse of the equivalent continuous growth rate?

I understand $e$ as $\lim_{n \to \infty} \big(1+\frac{1}{n}\big)^n$. I also (finally) understand the idea that continuous growth is "a rate that is applied constantly to the amount present at any ...
17
votes
4answers
1k views

Is this alternative notion of continuity in metric spaces weaker than, or equivalent to the usual one?

I will try to be as clear as possible. For simplicity I will assume that the function $f$ for which we define continuity at some point is real function of a real variable $f: \mathbb R \to \mathbb R$...
0
votes
1answer
64 views

Is $\cos^3{x}$ uniformly continuous?

I've found proofs that a continuous mapping from a bounded set to $\mathbb{R}$ is uniformly continuous, but is it true the other way? If not, how do you go about proving or disproving uniform ...
0
votes
2answers
28 views

Eliminate removable discontinuity

I have two rational functions which I have to examine for discontinuity and try to remove their domain gaps if possible. $$f(x)=\frac{|x|-1}{x^3-x} $$ and $$g(x)=\frac{sin (x)}{sin (2x)} $$ I ...
0
votes
2answers
43 views

Give an example of a space $X$, a subset $A$ and topology on $A$ such that $i: A \rightarrow X$ is NOT continuous

I was thinking of taking $X=\{0,1,2\}$ and $A=\{1,2\}$ And then the primage of ${0}$ is empty? I do not know which topology to take Thanks
1
vote
1answer
49 views

Continuity of the addition of two functions

We have seen that $(f + g)$ is continuous at $x = c$ whenever both $f$ and $g$ are. Determine whether the following statements are true or false. Justify your answers. (a) If $f$ is continuous at $x =...
2
votes
0answers
84 views

Continuous strictly increasing function is absolutely continuous iff set of infinite derivative maps to measure zero set

If $u:[a,b]\to\mathbb{R}$ is continuous and strictly increasing, prove that $u$ is absolutely continuous iff it maps $E:=\{x\in[a,b]:u'(x)=\infty\}$ into a set of measure 0. This question came from ...
1
vote
0answers
112 views

If $f:\mathbb R\to\mathbb R$ continuous does $f^{-1}$ also continuous?

Let $f:\mathbb R\to\mathbb R$ is bijective and continuous, does $f^{-1}$ is also continuous ? Does this result hold for $f:U\to\mathbb R$ where $U\subset \mathbb R^n$ ? and for $f:V\to\mathbb R^m$ ...
1
vote
1answer
41 views

Which of the following function(s) has/have removable discontinuity at $x=1?$

Which of the following function(s) has/have removable discontinuity at $x=1?$ $(A)f(x)=\frac{1}{\ln|x|}\hspace{1cm}(B)f(x)=\frac{x^2-1}{x^3-1}\hspace{1cm}(C)2^{-2^{\frac{1}{1-x}}}\hspace{1cm}(D)\frac{\...
1
vote
1answer
43 views

Rational Multivariable limit

I am having some issues with the following multivariable limit: $$\lim_{x,y\to0,0} \frac{x^2+y^2}{x+y}$$ I am trying to show whether it exists and is equal to 0, or whether it does not exist. What I ...
0
votes
0answers
37 views

Finding $\lim_{n\to \infty}\;n\theta(n)$

Let $f$ be a positive and continuous function in the interval $[0,1]$. It can be shown that there exists for every natural number $n$, a real number $\theta(n)\in [0,1]$ such that $$\frac{1}{n} \...