# Tagged Questions

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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### How to show that $f(x)=x^2$ is continuous at $x=1$? [closed]

How to show that $f(x)=x^2$ is continuous at $x=1$?
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### Proof of continuity of Thomae Function at irrationals.

In Thomae's Function: \begin{align} t(x) = \begin{cases} 0 & \text{if x is irrational}\\ \frac{1}{n} & \text{if x = \frac{m}{n} where \gcd(m,n) = 1} \end{cases} \end{align} I ...
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### Is there a monotonic function discontinuous over some dense set?

Problem (for fun--not homework) Can we construct a monotonic function $f : \mathbb{R} \to \mathbb{R}$ such that there is a dense set in some interval $(a,b)$ for which $f$ is discontinuous at all ...
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### Equicontinuity on a compact metric space turns pointwise to uniform convergence

I know that If $\{f_n\}$ is an equicontinuous sequence, defined on a compact metric space $K$, and for all $x$, $f_n(x)\rightarrow f(x)$, then $f_n\rightarrow f$ uniformly. I'm having trouble ...
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How do I prove $\sin x$ is uniformly continuous on $\mathbb R$ with delta and epsilon? I proved geometrically that $\sin x<x$ and thus, $$|f(x_1)-f(x_2)|=|\sin x_1 - \sin x_2|\le|\sin x_1|+|\sin ... 1answer 1k views ### Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous? Let (X,d) and (Y,\rho) be metric spaces and f:X\to Y be a function and suppose for any Cauchy sequence (a_n) in X, (f(a_n)) is a Cauchy sequence in Y. Is f continuous? Let f be ... 1answer 223 views ### f defined on [1,\infty ) is uniformly continuous. Then \exists M>0 s.t. \frac{|f(x)|}{x}\le M for x\ge 1. f defined on [1,\infty ) is uniformly continuous. Then \exists M>0 s.t. \frac{|f(x)|}{x}\le M for x\ge 1. I know f uniformly continuous \implies \forall\varepsilon > 0s.t.\forall x,y\... 5answers 1k views ### Is a rational-valued continuous function f\colon[0,1]\to\mathbb{R} constant? Let f\colon[0,1]\to\mathbb{R} be continuous such that f(x)\in\mathbb{Q} for any x\in[0,1]. Intuitively I feel that f is constant, since \mathbb{Q} is dense in \mathbb{R}. How can I ... 2answers 527 views ### f brings convergent nets to convergent nets, is it continuous? Let f:(X,\mathcal T)\to (Y,\mathcal S) be a function between topological spaces. Let for any convergent net (x_\alpha) in X, (f(x_\alpha )) be convergent in Y. Is f continuous? (It seems ... 1answer 2k views ### Continuity of a convex function I'm trying to solve the following problem: Let f:K\rightarrow \mathbb{R} , f convex and K \subseteq \mathbb{R}^n convex. Then f is continuous on K. I have proved the only case n=1, but ... 2answers 1k views ### For a continuous function f and a convergent sequence x_n, lim_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n) [duplicate] Let f:X \rightarrow Y be a function. Prove that if f is continuous, then for every convergent sequence (x_n) lim_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n) My ... 2answers 2k views ### f,g continuous from X to Y. if they are agree on a dense set A of X then they agree on X Problem: Suppose f and g are two continuous functions such that f: X \to Y  and g : X \to Y . Y is a a Hausdorff space. Suppose f(x) = g(x)  for all x \in A \subseteq X  where A ... 3answers 832 views ### sequentially continuous on a non first-countable Can you give me an example of a function which is sequentially continuous but not continuous? (I know that in first-countable spaces this is equivalent, but what about in spaces without this condition?... 4answers 4k views ### If f,g are uniformly continuous prove f+g is uniformly continuous Suppose f:E \rightarrow \mathbb{R} and g:E \rightarrow \mathbb{R} are uniformly continuous, where E is a subset of \mathbb{R}. Show that f+g is uniformly continuous. What about fg and \... 2answers 115 views ### Assuming: \forall x \in [0,1]:f(x) > x Prove: \forall x \in [0,1]:f(x) > x + \varepsilon  Let f a continous function defined in the interval [0,1]. Assuming: \forall x \in [0,1]:f(x) > x Prove: \forall x \in [0,1]:f(x) > x + \varepsilon  I tried to use Heine–Cantor theorem ... 1answer 562 views ### Can a continuous function from the reals to the reals assume each value an even number of times? Suppose f: \mathbb{R} \rightarrow \mathbb{R} is continuous. Is it possible for f to assume each value in its range an even number of times? To clarify, some values might be taken 0 times, some 2, ... 1answer 400 views ### Continuous map f : \mathbb{R}^2\rightarrow \mathbb{R} Let f : \mathbb{R}^2\rightarrow \mathbb{R} be a continuous map such that f(x)=0 only for finitely many values of x. Which of the following is true? Either f(x)\leq 0 for all x or f(x)\... 1answer 1k views ### A continuous bijection f:\mathbb{R}\to \mathbb{R} is an homeomorphism? A continuous bijection f:\mathbb{R}\to \mathbb{R} is an homeomorphism. With the usual metric structure. I always heard that this fact is true, but anyone shows to me a proof, and I can't prove it. ... 3answers 6k views ### Why are norms continuous? Describe why norms are continuous function by mathematical symbols. 1answer 2k views ### Is the total variation function uniform continuous or continuous? I have been doing some excercises on total variation when the following questions came up to my mind: (1) Let f be continuous on the interval [0,1] and be of bounded variation. Is it true that ... 7answers 2k views ### How to prove continuity of e^x. I simply want a proof that e^x is continuous. I have never really been able to find something satisfying these points: e is defined to be the limit \lim_{n\to\infty}\left(1+{1\over n}\right)^n... 2answers 206 views ### Continuous function positive at a point is positive in a neighborhood of that point Pretty much the problem asks if a function is continuous at the point c and f(c) > 0 then there exists a d > 0 such that \forall x, f(x) > 0 with |x-c| < d. I can understand ... 3answers 1k views ### A function takes every function value twice - proof it is not continuous I want to prove the following nice statement I've found: A function f: [0,1] \rightarrow \mathbb{R} takes every function value twice - proof it is not continuous I've already found an answer to my ... 1answer 1k views ### Show that f is uniformly continuous if limit exists Let f(x) be continuous on (0,1]. Show that f is uniformly continuous IFF \displaystyle \lim_{x\to0^+} f(x) exists. Thoughts: Backward Proof: Let another function \overline f(x) be ... 1answer 626 views ### Continuity of absolute value of a function Let f(x) be a continuous function. Prove that \left|f(x)\right| is also continuous. Is it correct to say that, by the reverse triangle inequality, \left|f(x)-f(c)\right| \geq \left|f(x)\right|-\... 3answers 127 views ### Continuity of \frac{x^3y^2}{x^4+y^4} at (0,0)? [duplicate] Suppose a function f is defined as follows:$$f(x,y)=\begin{cases} \frac{x^3y^2}{x^4+y^4}&\text{ when }(x,y)\neq(0,0),\\0 & \text{ when }(x,y)=(0,0).\end{cases}$$Is this function ... 1answer 466 views ### |f(x)-f(y)|\geq k|x-y|.Then f is bijective and its inverse is continuous. My exercise says: Let f:\mathbb{R} \rightarrow \mathbb{R} a continuous function e suppose that exists k such that:$$|f(x)-f(y)|\geq k|x-y|$$Then f is bijective and its inverse is continuous.... 3answers 2k views ### Does \lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0 imply that f is continuous? Suppose f is a real function defined on \mathbb{R} which satisfies$$\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0.$$Does this imply that f is continuous? Source: W. Rudin, Principles of ... 3answers 133 views ### Example of topological spaces where sequential continuity does not imply continuity Please give an example of a function f : X \to Y  where X,Y are topological space , such that there exist x \in X such that for every sequence \{x_n\} in X converging to x , \{f(x_n)\} ... 1answer 168 views ### a continuous mapping is determined by its values on a dense set Let f and g be continuous mappings of a metric space X into a metric space Y and let E be a dense subset of X. Prove that f(E) is dense in f(X). If g(p)=f(p) for all p \in E, prove ... 2answers 1k views ### Is every convex function on an open interval continuous? Let f:(a,b)\rightarrow \mathbb{R}. f satisfied the following property: If \forall x_{1},x_{0},x_{2}\in(a,b) and x_{1}<x_{0}<x_{2};then\frac{f(x_{0})-f(x_{1})}{x_{0}-x_{1}}\geq \frac{... 1answer 2k views ### Prove continuity for cubic root using epsilon-delta I am trying to prove that a function is continuous at a point a using the \epsilon-\delta theorem. I managed to find a \delta in this case |2x^2+1 - (2a^2+1)| < \epsilon. But I have a hard ... 3answers 472 views ### Fixed point in a continuous function Suppose that f is a function defined in [a;b] to [a;b] and continuous on [a;b]. The problem is I haven't the definition of the function, this is more abstract, but even if how can I prove that ... 1answer 86 views ### Prove symmetry of probabilities given random variables are iid and have continuous cdf Let Y_1, Y_2, ... be independent and identically distributed random variables in (\Omega, \mathscr{F}, \mathbb{P}) s.t. their distributions are continuous and$$F_{Y}(y) := F_{Y_1}(y) = F_{Y_2}(y) ...
Let $S = [0,1) \cup [2,3]$ and $f\colon S \rightarrow \mathbb R$ be such that $f(S)$ is connected . Which of the following are true: a) $f$ is discontinuous exactly at one point. b) $f$ is ...