Questions tagged [continuity]

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) or (general-topology)

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Let f : R → R be continuous, such that lim f(x), x→−∞ and lim f(x), x→+∞ exist and are real numbers. Prove that f is uniformly continuous

For f as x tends to -∞ , I have this: Consider ε > 0. Because f converges to a limit in −∞, we know that given ε> 0, there exists an M such that for all x < M, |f(x)-L| <= ε/2, (where lim ...
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If $f$ is differentiable in a neighbourhood of $c$, is $f'(x)$ continuous at $x=c$?

I'm pretty sure the statement "If $f$ is differentiable at $c$, is $f'(x)$ continuous at $x=c$" is wrong and there are quite a lot of counterexamples to that, but what if we add a " ...
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Domain and Continuity of piecewise composite function

Consider the function $$g(x)=\begin{cases} e^{-x}& \text{ if } x\geq 0 \\ \sqrt{\left | x \right |}& \text{ if } x< 0 \end{cases}$$ and $$h(x)=(x-4)(x+1)^2$$, when is the composite ...
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1answer
59 views

Proving a f is continuous at 1/3

Show that $f(x) = \frac{1}{5x}$ is continuous at $x = \frac{1}{3}$. I have to use an $\varepsilon - \delta$ proof? I am having trouble choosing my delta because I am confused as to how $|\frac{1}{x} - ...
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25 views

How To Prove This Is Continuous Or Discontinuous

How do I prove this is continuous/discontinuous? A function $f(x)$ (can be any function, in general) is defined on the interval $[3,7]$, where $f(3)=2$ and $f(7)=4$. The range of $f(x)$ on this ...
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A seemingly-complicated question about derivatives and continuity.

Question : My problem lies in the last question. I think I have managed to solve (a), (b), and (c). I got $$f(0)=f'(0)=A=g'(0)=0$$ and $$g'(x)=f'(x)\sin\left(\frac{1}x\right)-\frac{f(x)\cos\left(\...
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30 views

prove that a composite function is continuous

For these functions: $$h(x)= \begin{cases} e^{-x} & x\geq 0 \\ \sqrt{|x|} & x<0 \\ \end{cases}$$ $$g(x)=(x−4)(x+1)^{2}$$ So obviously h(x) is discontinuous at x =0, but how ...
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24 views

Limit and continuity application [closed]

Let $f$ be a real-valued continuous function such that $\displaystyle\lim_{x \to 0}f(x)\cos^2(\frac{\pi}{x})=0.$ Find $f(0)$. Since $f$ is continuous, $\displaystyle\lim_{x \to 0}f(x)=f(0)$. I guess ...
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23 views

Why can't set of discontinous points with volume zero intersect with the boundary of a rectangle?

In Multivariable Mathematics by Dr. Shifrin, Prop. 1.8 in chapter 7 says Suppose $f:R\to \mathbb{R}$ is a bounded function and the set $X=\{\mathbf{x}\in R:f \text{ is not continuous at }\mathbf{x}\}$ ...
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Counterexample of Continuity of a function

If a function $g(x)$ has a domain [4,8] with and a range of [2,4], with $g(4)=2$ and $g(8)=4$. Furthermore, assume that g only achieves each value on [2,4] exactly once. In other words, for each y in ...
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Open neighborhood of a subset of a metric space

Suppose $X$ is a metric space, with distance function $d:X\times X\to \Bbb R$. Also suppose $U$ is an open neighborhood of a subset $A\subset X$. Then for each $a\in A$, we can choose an $\epsilon_a&...
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27 views

Continuity of $(x,y)\mapsto(y,x)$ [closed]

Is this map continuous for real spaces $X,Y$? I can’t see why it wouldn’t be but at the same time I can’t seem to come up with concrete reasoning why it would be true. Thanks in advance!
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How to check if a function is continuous on some interval?

What the methods exists for check "Jump discontinuity" on some interval? I am interested in numerical methods because I need to implement this in a high-level, general-purpose programming ...
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1answer
26 views

Proving a function is differentiable iff it's differentiable at a point

In my calculus notes I found the following exercise: Suppose that $f:(0,\infty)\to\mathbb{R}$ satisfies $f(x)-f(y)=f(x/y)$ for every $x,y\in(0,\infty)$ and $f(1)=0.$ (a) Show that $f$ is continuous ...
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59 views

Differentiable at $x=a$ implies continuous at $x=a$

Consider the function $$f(x)=\left\{\begin{array}{cc} x^2-4 & \text{ if }x\leq 2\\4x+3&\text{ if } x\gt 2\end{array}\right.$$ This function is differentiable at $x=2$ since $\lim_{h\to 0^{\pm}}...
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Someone help me with this one :(( [closed]

For what values of a is the function f(x)={ax+5, x<4 x²-x, x≥4 continuous at x=4? a.7/4. b. -7/4 c.0 d.-5
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If $f$ is defined on $R$ and $f$ is continuous at $x=0$ and $x=1$, prove that, if $f(x^2)=f(x)$, then $f$ is a constant function $\forall x\in R$

It's all stated in the title, actually; to relate to the continuity, I probably have to use the $\epsilon-\delta$ definition, and work some magic with the $f(x^2)=f(x)$, but that's getting me nowhere--...
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Proving function $f(x)=\frac{1}{\sqrt{1+x^2}}$ continuous using the definition of sequential continuity

My goal is to prove $f(x)=\frac{1}{\sqrt{1+x^2}}$ is continuous. I think it will be easier to approach this using the def of sequential continuity? Proof: Prove $\left\{a_{n}\right\}$ converges to a ...
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29 views

show that $ f $ is continuous in $a$

If $I = [a,b] $ and suppose $f : I → R $ is increasing in $I$, then $f$ is continuous in $a$ if and only if $f(a)$ = $inf$ $\{f(x):x ∈→ (a,b)\}$ I've solved it this way, okay? If not, please help me ...
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32 views

Comparison principle for differential equations

I am trying to solve example 3.8 in the book Nonlinear systems by Hassan Khalil and I have been unable to figure out how they got the answer for $\frac{\mathrm dv(t)}{\mathrm dt}$ as $-2x^2(t)$. I ...
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1answer
42 views

Is continuous a function if and only if preserves order? [closed]

I'm trying to prove that. Let $(X,\leq)$ a partial ordered set, with the associated topology formed by the subsets $U$: $ x \in U$ and $ y \leq x \Rightarrow y \in U$ Then a function $f$ is continuous ...
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1answer
29 views

Showing that a set is open in the strong topology induced by a family of functions

Let $Y$ be an arbitrary set. There is a natural way in which a family of maps $f_i: X_i → Y$ (for $i ∈ I$) from topological spaces $X_i$ to $Y$ induces a topology on $Y$. Namely, the strong topology ...
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1answer
19 views

find linear approximation

given function $f:\mathbb{R^2} \to \mathbb{R^2}$ $f(x,y)=(x^3y^2-y,xy^3-x)$ how do we calculate linear approximation to $(fof)(1+h,1+k)$ for $h,k$ near $0$ i got this $f(x_o+ \triangle x,y_o+\...
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The proof of the product of two continuous functions is also a continuous function in multivariable case.

I want to show that the the product rule of continuous function can be extend to the multivariable case. So, by my setting, given two functions $f(x)$ and $g(y)$ are both continuous function defined ...
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1answer
21 views

Are the product, quotient, and composition property of two continuous function hold for two functions with respect to different variables?

For continuous function in single variable case, we have following: if $f(x)$ and $g(x)$ are continuous function, then: $$h(x)=f(x)⋅g(y)\; is\; also\; a\; continuous\; function.$$ $$h(x)=\frac{f(x)}{...
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2answers
36 views

Existence of continuous function on $[0,1]\cup[2,3]$ [closed]

Is there exist continuous function $f:[0,1]\cup[2,3]\to \{0,1\}$
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1answer
24 views

Convergence of a CDF to another CDF

Suppose $X$ and $Y$ are two independent random variables. Further, suppose that $Z := X + cY$ is also a random variable for all constants $c > 0$. If we assume that the corresponding cdfs $F_X$ and ...
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Finding torque and power using angular acceleration [closed]

How to extract the angular acceleration value from continuity equation [1]: https://i.stack.imgur.com/KJ3h5.jpg
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46 views

Showing that a given function is continuous at all irrational points and discontinuous at all rational points

So, here's the problem: Let $(r_n)_{n=1}^{\infty}$ be an enumeration of $\mathbb{Q}$. Then, define a function: $$f(x) = \begin{cases} \frac{1}{n} & x = r_n \\ 0 & x \in \mathbb{R} ...
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81 views

Continuity on an open interval and the epsilon-delta definition of continuity

The question asks: Assume $f$ is continuous at $a$. Assume $f(a) < 4$. Prove there exists an open interval $I$, centred at $a$, such that $\forall x \in I, f(x) < 4$. I am stuck on this question....
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0answers
21 views

Finding that no vector in prehilbert space has a certain property

Let $E=C^0([0,1],\mathbb C)$ with the norm derived from the inner product: $$ <f,g>=\int_0^1 f(t)\overline {g(t)}dt $$ I had to show that for the continuous linear transformation, $u:E\to\...
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40 views

showing that $f$ is not continuous at $(0,0)$.

Show that $$f(x,y)=\begin{cases}\frac{x^2+ y^2}{x-y} & x \ne y \\ 1 & x=y \end{cases}$$ is not continuous at $(0,0)$. I was trying this question by showing that the limit of the function at $(...
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1answer
26 views

Question regarding Thomae's function continuity proof

The function defined by $$f(x)=\begin{cases} \frac{1}{q} & x=\frac{p}{q}\in\mathbb{Q}, (p,q)=1 \\ 0 & x\in\mathbb{I} \end{cases}$$ is continuous at every irrational in $]0,+\...
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19 views

Existence of uniformly continuous function such that square is not uniformly continuous on closed interval

$f:[-1,1]\rightarrow \mathbb{R}$ is uniformly continuous, is it possible that $f(x)^2$ not uniformly continuous. I suppose no since $f(x)$ is continuous and so is $f(x)^2$ so $f(x)^2$ must be ...
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How I check continuity and differentiability of a map defined from $f:\Bbb{R}^2 \to\Bbb{R}^2$

Consider the map $f:\Bbb{R}^2 \to\Bbb{R}^2$ defined by $f(x,y)=(3x-2y+x^2 , 4x+5y+y^2)$. Then how I find continuity and differentiability and directional derivative at $(0,0)$?
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2answers
25 views

Continous Piecewise Function [closed]

I recently found this question: find $c$ and $d$ such that $$f(x) = \begin{cases} cx+4d & x<2\\ x^{2}+4 & 2\leq x\leq 3 \\ dx^{2}+\frac{2x}{c}+1 & x>3 \end{...
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1answer
28 views

Find a function $f$ such that $f$ is discontinuous but $W(f,x)$ is a continuous function

I need some help Find a function $f$ such that $f$ is discontinuous but $W(f,x)$ is a continuous function (where $W(f,x)$ is the oscillation of f in a point, this means that $W(f,x)=inf[M_\delta(f)-m_\...
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1answer
19 views

Continuous and one-one function that is bounded

Actually option 2 is given as answer.So,other options are incorrect.I got an counter example for other options that they are not true.But I am unable to find a counter example to show option 4 is ...
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2answers
36 views

Continuity of the derivative function. [closed]

Suppose we have a function f(x) that is differentiable for all values of x. Is it necessary for the derivative function to be continuous for all x ?.
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1answer
37 views

Show that if $f_n$ converges uniformly on $[0,1]\cap\mathbb{Q}$, then $f_n$ converges uniformly on $[0,1]$.

The question is as follows Let $f_n$ be a sequence of continuous functions defined on $[0,1]$. Suppose that $f_n$ converges uniformly on $[0,1]\cap \mathbb{Q}$ to $f$. Show that $f_n$ converges ...
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13 views

Elimination of a candidate of optimal solution based on SOC

A function $f(x; \alpha)$ is continuous in both variable $x$ and parameter $\alpha$,first and second partial derivatives of $f$ are continuous as well. I want to find the optimal solution of the ...
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1answer
44 views

Function discontinuous on $(0,1)$ continuous everywhere else

Find a function continuous everywhere but $(0,1)$. Attempt: Let $$f(x) = \begin{cases} -x+1,& \quad x<0\\1, & \quad x \in \mathbb{Q}\cap [0,1] \\ 0, & \quad x \in \mathbb{Q^c}\cap [0,1] ...
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Struggling with basic limit problem

As part of some other proof, I'm trying to show the following: Let $A \subset \mathbb{R}^n$ and $a \in \overline{A}-A$ (ie. in the closure of A but not in A) so that the function $f : A \to \mathbb{R}$...
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1answer
54 views

Is this Integral always continuous?

PROBLEM: Let $K:[a,b]\times [a,b]\to\mathbb{R}$ y $f:[a,b]\to\mathbb{R}$ continuous functions. Let $\hat{f}:[a,b]\to\mathbb{R}$ given by: \begin{equation*} \widehat{f}(x):=\int_{a}^{b}K(x,y)f(y)...
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1answer
98 views

How to prove $\mathop {\lim }\limits_{x \to {\rm{ + }}\infty } f'(x){\rm{ = 0}}$

Today I have came across a problem.It is harder than I think. Given that (1) $f(x)$ is differentiable on $\left[ {{\rm{0}},{\rm{ + }}\infty } \right)$, (2) $f'(x)$ is uniformly continuous on $\left[ ...
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0answers
10 views

When is the continuous extension of an elementary function elementary?

Some continuous extensions of elementary functions are elementary. For example $f(x) = \frac{x}{x}$ is elementary in $\mathbb{R}\setminus\{0\}$ and its continuous extension to $\mathbb{R}$ is $g(x)=1$ ...
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1answer
39 views

Proof of convergence and $C^1$ class $\sum_{n=1}^{\infty}e^{-n^2x}$

I have a problem to prove that this function: $$\sum_{n=1}^{\infty}e^{-n^2x}$$ takes every value in its domain and is $C^1$ class in $x\in(0,\infty)$. The steps I have to take is to check the ...
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33 views

Continuity of a function at points where it became indeterminate form

If a function $f(x)$ at point let's say at $x=a$, $f(a)$ is an indeterminate($\frac{0}{0}$ etc. ) form then 1.is it sufficient to say that function will be discontinuous at $x=a$ Even though limit ...
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2answers
55 views

Product topology and continuity

Hi guys I'm new to topology and was asked to prove the following, of which I am having troubles with: Let $F:X \times I \rightarrow Y$ be a continous function. For each $t \in I$ let $f_{t} = F(x,t)$. ...
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15 views

A question in Topology- the likage between a continuous function and a convergent sequence.

Let $f:X\to Y$ be a continuous function (where $X$ and $Y$ are topological spaces). Let $\lim_{n\to\infty}x_n=x$. I wanted to prove that $f(x_n)$ converges to $f(x)$ as $n$ goes to infinity. Here is ...

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