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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Monotone functions and distribution functions

I found this quote in a textbook on measure theory I'm studying: Let $f:[a,b] \to \mathbb{R}$ be an increasing function. Since $f$ has only countably many discontinuities, we may assume without ...
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Can a real continuous bounded function on $ \Bbb{R}^{2} $ be expressed as a finite sum of products of real continuous functions on $ \Bbb{R} $?

Can a real-valued continuous bounded function on $ \Bbb{R}^{2} $ always be expressed as a finite sum of products of real-valued continuous functions on $ \Bbb{R} $?
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Characteristic function approximated by continuous function

I am trying to do the following problem Let $E \subset \mathbb R^d$ be measurable and let $\epsilon>0$. Show that if $A \subset E$ is measurable, then there is $f:E \to \mathbb R$ continuous such ...
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“root” of a right-continuous function

Suppose $f:[0,1] \longrightarrow [-1,1]$ is a right-continuous function such that $f(0) < 0$, $f(1) > 0$, and $f$ only changes sign once in the interval $[0,1]$. Suppose we define the "root" of ...
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29 views

Continuity and Directional Derivatives

Does every absolutely continuous function on a compact set possess a left and right hand derivative everywhere on its interior? Although the two need not be equal of course.
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Does the function $f(x) = \frac{x^2-1}{x-1}$ have any point discontinuity?

Since the domain of $f(x)$ is $(-\infty, 1) \cup (1, \infty)$ is there any point discontinuity in $f(x)= \frac{x^2-1}{x-1}$?
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What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference via geometrically? What is the best way to describe the difference between these two concepts to someone else? Where the ...
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Show that $f$ is continuous at exactly one point

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $$f(x)= \begin{cases} 5x+7 & \text{ if } x \text{ is rational } \\ x+11 & \text{ if } x \text{ is irrational } \end{cases}$$ ...
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If $f(U)=0$ then what is possible?

Let , $U=\left(0,\frac{1}{2}\right)\times \left(0,\frac{1}{2}\right)$ and $V=\left(-\frac{1}{2},0\right)\times \left(-\frac{1}{2},0\right)$ and $D$ be the open unit disk centered at origin of $\mathbb ...
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2answers
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Continuous function without a weak derivative

Let $f:\Omega\to\mathbb{R}$ be a continuous function. Is it necessarily true that $f$ has a derivative in the weak sense? That is, is there some $v:\Omega\to\mathbb{R}$ such that for every test ...
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continuity and convergence [duplicate]

If we have a continuous function that converges on a compact subset of a metric space does it imply that it converges uniformly in general, or is this only in the case if f is monotonic (Dini's ...
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Convergence of a sequence of integrals

I've tried expanding the hinted expression by using the definition from part (i) and choosing an X0 sufficiently large that |f(x)-l| < 1 but this doesn't appear to help very much at all. I've ...
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Continuity and measurability

this question concerns continuity and measureability. Am I right in thinking that if $f>0$ for all x then $\log(f(\lambda x)/f(x))$ is a continuous function for all $x$. Does this then mean that ...
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Continuity of $\mu(t)=\inf\{x \in \mathcal C : \kappa(x)=t\}$.

Let $\Delta = \{ 0, 1\}^{\mathbb N}$ be a Cantor set. Define $\theta : \Delta \to [0,1]$ by the formula $$\theta(x_1,x_2,\dots) = \sum_{n=1}^\infty \frac{2x_n}{3^n}.$$ Denote $\mathcal C = ...
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Continuity of a function in a locally convex topological space

I endow the space of bounded sequences with a locally convex topology $\tau$ such that $\tau$ is strictly finer than the product topology (the topology of pointwise convergence), $\tau_p$, and ...
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Showing that the functional $L[h(x)]=\int_{a}^{b}h(x)f(x)dx$ is continuous

Suppose that we have the functional $L: L^2[a,b] \to \mathbb{R}$ , $L[h(x)]=\int_{a}^{b}h(x)f(x)dx$. $f(x)$ is a well behaving, integrable function in $L^2[a,b]$. I want to show that this is a linear ...
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Can unbounded discontinuous functions be locally bounded?

Consider the function $$f(x) = \frac{x^3}{1+x^3}$$ Obviously this function is discontinuous at $x = -1$ therefore discontinuous on $\mathbb{R}$. Moreover, it is unbounded at the same point. Now, I ...
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38 views

Two functions equal in some point

I have two continuous functions $f,g$, $f(0) \lt g(0), f(1) \gt g(1)$. How do I prove without using "advanced" theorem (using only definitions of limit, continious functions and sup/inf definitions), ...
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31 views

Rationale behind a proof regarding a continuous function and an open ball

can I have the rationale for the first line of this proof? i.e. How did you know to start answering the question in this manner? I am guessing it is because you want to exploit the definition of ...
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22 views

On the definition of uniform continuity over an interval.

I was reading some slides and I stumbled upon this definition of uniform continuity in an interval I am unsure on how to trace this back to the definition of uniform continuity that I know: A ...
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Proving isometry and continuity from a positive definite symmetric real matrix

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\epsilon$ be the Euclidean metric on $\Bbb R^n$ ...
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Surjectivity of expanding map

Suppose that $(X, d)$ is a compact metric space and that $f: X \rightarrow X$ is a continuous function satisfying $d(x,y) \leq d(f(x), f(y))$ for all $x, y \in X$. Show that $f(X) = X$. Here is a ...
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Proving the continuity of functions from one metric to another

I'm studying mathematics at university, and am having trouble with some of the continuity questions. The following is a question from a previous assignment that I was unable to complete. The original ...
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39 views

Continuity of a peculiar function

I came across the question: Evaluate$f(x)=\lim_{m \to \infty}\lim_{n \to \infty}[\cos(n!\pi x)]^{2m}$. I simplified this to: $$f(x)= \begin{cases} 1 & \text{if $x \in Q$} \\ 0 & \text{if ...
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If function has a given limit then to prove that function is bounded.

How to Prove that if a function $f : A \to \Bbb R$ has a limit $l \in \Bbb R$ at $c \in L(A)$, then it is bounded in a neighborhood of $c$, i.e. there exists $M \in \Bbb R$ and $\delta > 0$ such ...
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Let $f:[0,\frac{\pi}{2}]\to R$ be $f(x)=\max\{x^2,\cos x\}$.Prove $f(x)$ attains minimum at $x_0$ and is a sulution to $x^2=\cos x$

I try to write $f(x)=\frac{1}{2}x^2+\frac{1}{2}\cos x+\frac{1}{2}|x^2-\cos x|$ and use the Extreme Value Theorem to show that $x_0$ exists in $\left[0,\frac{\pi}{2}\right]$, but I don't know how to ...
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Boundedness Theorem for continuous functions on intervals

Just want to confirm this is a suitable proof: Assume $f$ is not bounded on $I$. So, for any $n \in \mathbb{N}$, $\lvert f(x)\rvert > n$. Since $I$ is bounded, $x_n$ is also bounded. By ...
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$f(x) =\ln(2x^2 + 1)$ is continuous on $\mathbb{R}$

True or False The function $f : \Bbb R \to \Bbb R$ defined by $f(x) = \ln(2x^2 + 1)$ is continuous on $\Bbb R$. I know this condition that The function $f$ is continuous at some point $c$ of its ...
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Let $f,g$ be continuous from $\mathbb R$ to $\mathbb R$ [duplicate]

Let $f, g$ be continuous from $\mathbb R$ to $\mathbb R$, and suppose that $f(r) = g(r)$ for all rational numbers $r$. Is it true that $f(x) = g(x)$ for all $x \in \mathbb R$?
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I need help finishing this proof using the Intermediate Value Theorem?

Let $f$ and $g$ be continuous functions on $[a,b]$ such that $f(a)\geq g(a)$ and $f(b) \leq g(b)$. Prove $f(x_0)=g(x_0)$ for at least one $x_0$ in $[a,b]$. Here's what I have so far: Let $h$ be a ...
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Test for uniform continuity

Test for uniform continuity the function $ f(x, y) = (x^2 + y^2)^\alpha \sin{\frac{1}{x^2+y^2}} $ in $ \{ x^2+y^2 > 1\} $ If we consider $ \alpha < 1 $, then $ \lim_{\sqrt{x^2+y^2} \to ...
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Expanding a function

Is it possible to expand a function $$ f(x,y) = \dfrac{\sin (xy)}{\sqrt{x^2 + y^2}} $$ so it will be continuous on $\mathbb{R}^2$? Now, the denominator should not be equal to $0$, so for the domain, ...
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Show that if $g((x_n)) \rightarrow l$ and $g((y_n)) \rightarrow m$, then $l=m$

Suppose that $g: (a,b] \rightarrow \mathbb{R}$ is uniformly continuous. Suppose that both $(x_n), (y_n)$ are sequence in $(a,b]$ which converge to $a$. Show that if $g(x_n)) \rightarrow l$ and ...
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Continuous multivariable function without limits in a point

I am curious, if there can be a function $f(x,y)$, which is continuous in a point $[0,0]$, but for which iterated limit $\lim _{x \to 0} \lim _{y \to 0} (f(x,y))$ does not exist. Is it even possible ...
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Show that $f$ is uniformly continuous.

Suppose that $f:(a,b] \rightarrow \mathbb{R}$ is continuous and that the limit as $\lim\limits_{x \rightarrow a}f(x)$ exists. Show that $f$ is uniformly continuous. I am really struggling with this ...
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Counter Example to Tietze Extension Property for Arbitrary Topological Space

Above is my question. My only issue is the final bit! For statements $1.$ and $2.$, the answer is true, since in both cases $Y$ is normal and we know that both metric and compact, Hausdorff spaces ...
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Need Help look at function continuity

Consider the following piece-wise function: $$f(x, y) = xy \frac{x^2-y^2}{x^2+y^2}$$ for $(x,y) \neq (0,0)$ and $f(0,0)= 0$. Discuss the continuity of $f$ at $(0, 0)$. Calculate $\partial f / ...
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A little guidance on finding the limit

How do I find the limit of $f(z) = \frac{x^2y}{x^3+y^3} + ixy$ as $z \to0$ ? What I think is if $z\to0$, that implies $x ,y\to0$. But since the $f(z)$ has both variables $x$ and $y$ mixed together, ...
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Proving the continuity of functions from metrics

Context: I'm studying mathematics at university, and am having trouble with some of the continuity questions. The following is a question from a previous assignment that I was unable to complete. The ...
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Finding the limit of complex function

I am trying to check the continuity of this complex function at the origin. $f(z)=\begin{cases} \operatorname{Im}( \frac{z}{1+|z|} ) \qquad &\mbox{when } z\neq0,\\ 0 \qquad ...
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Real Analysis: Continuity

$f(x)=\left\{ x^2+x, x \in \Bbb Q\right\}, f(x)=\left\{ x^3 + 1, x \notin \Bbb Q \right\}$ I want to prove that $f$ is discontinuous at $x \ne 1$. What I have so far is: Fix $\delta > 0$. We ...
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finding the continuity of a function

I need to find the value of $a$ for which the function $f(x,y)= \frac{x^2-y^2}{x^2+y^2}$ if $(x,y) \neq (0,0) $ and $f(x,y)=a$ when $(x,y)=(0,0)$ when continuous along the path $y=b\sqrt{x}$ where ...
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Need help in understanding proof of continuity of monotone function

I am reading the following proof of a proposition from Royden+Fitzpatrick, 4th edition, and need help in understanding the last half of the proof. (My comments in italics.) Proposition: Let $A$ be ...
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a continuous function

Let $C([a,b])$ be the collection of all functions $f:\mathbb{R} \to \mathbb{R}$ such that continuous on $[a,b]$. It is known that if $f\in C([a,b])$ then $f$ is continuous on every sub-interval of ...
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If a continuous function is strictly decreasing before a point and strictly increasing afterwards, is the point a global minimum?

I'm in the middle of a proof that a point on a function is a global minimum. Usually I'd just solve an inequality to prove by contradiction that there are no points less than the minimum. But I can't ...
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Continuity of the joint distribution function given continuity of marginals

Suppose $X$ and $Y$ are continuous random variables such that $F_X$ and $F_Y$ are the respective distribution functions. Suppose $F_X$ is continuous at $x_0$ and $F_Y$ is continuous at $y_0$. Then ...
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Smooth function conditions

A curve defined by $x=f(t)$, $y=g(t)$ is smooth if $f′(x)$ and $g′(x)$ are continuous and not simultaneously zero. Why do we have the second condition(simultaneously zero)?
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Is $f\colon Y'\to Y$ continuous?

Consider $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and $T\colon X\to X$ continuous, describing the following dynamics: For $\eta\in X$ let $\eta(y)$ describe the y-th position in the bi-infite sequence ...
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Help Understanding Gradients

I understand that gradients are vectors with partial derivatives as components when working in 3D space, but does the the existence of a gradient at a point imply continuity at that point?
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Is any compact, path-connected subset of $\mathbb{R}^n$ the continuous image of $[0,1]$?

If $f:[0,1] \to \mathbb{R}^n$ is any continuous map, then the image $f([0,1])$ is a compact, path-connected set, which is easy to show using some elementary topology. My question is the converse: ...