1
vote
0answers
25 views

Question about writing a proof with continuous functions [duplicate]

How would I write a proof for this example? We know that all polynomial functions on the reals are continuous by using the sequential definition of continuity. In particular, we know that the ...
7
votes
1answer
517 views

Where is the error in my proof that all derivatives are continuous?

I know that this can not be true due to counter-examples but I don't know where the error in my reasoning is. Assumption: If $f(x)$ is differentiable in $\mathbb{R}$ then the derivative $f'(x)$ is ...
0
votes
1answer
35 views

(Dis)continuity of function in $R^2$

$$f(x,y) = \begin{cases} a+2x^{2}-b(y-c), & x^{2}>2+x\wedge y<6\\ 3+cx-y, & else \end{cases}$$ $f(x,y)$ is continuous on $R^2$ if $a=-3, b=1, c=2$ I think it's true: insert ...
4
votes
4answers
110 views

If $f(x)$ is discontinuous at $x=0$, can $\int_{-1}^1 f(x)dx$ exist.

I am interested in the reasoning. All help is appreciated
1
vote
1answer
26 views

Well-Posedness PDE of the Form $\partial_t u = P(\partial_x) u$ for a Polynomial $P$

My question is to determine whether the PDE $\partial_t u = P(\partial_x) u$, with $2\pi$-periodic boundary conditions, for a polynomial $P$, is well-posed; this depends on the polynomial, and my ...
1
vote
1answer
41 views

Requirements for integration by parts/ Divergence theorem

In order to use the integration by parts formula(or more generally the divergence theorem) for functions of several variables $$\int_{\Omega} \nabla u\cdot v d \Omega = \int_{\partial \Omega}(u(v ...
2
votes
2answers
48 views

Requirement for continuity of unit normal vector

When considering a subset $\Omega \subset \mathbb{R}^{n}$. If we consider $\nu$, the outward unit surface normal to $\partial \Omega$, what are the requirements of $\partial \Omega$ which will ...
0
votes
1answer
41 views

Continuity basic understanding

I have been asked to figure out if they are continuos or discontinues or left or right con/discon for the point -2. -1. 0. 1. 2. , where the function g(x) has domian[-2,2]. I just do not get it. As ...
1
vote
3answers
56 views

True or False Question About Functions [closed]

If $f(1)>0$ and $f(3)<0$, then there exists a number $c$ between $1$ and $3$ such that $f(c)=0$. I'm not sure how to solve this question. Thanks in advanced!
0
votes
2answers
31 views

Determine intervals on which s(t) =equation

Determine the intervals on which $$s(t) = \frac{|t^2-2t - 3|}{t + 1}$$ is continuous. Hint: Use continuity checklist and check left and right continuity of proposed intervals which include ...
0
votes
0answers
37 views

L'Hospital's rule for higher derivatives

Let $u,v \in C^\infty(\mathbb{R})$, where $u(0) = 0$ and $v(0) = 0$ and $v'(0) \not= 0$. Then, one can define a function $f \in C^\infty(\mathbb{R}\setminus\{0\})$ by $f := u/v$. L'Hospital allows ...
1
vote
0answers
25 views

Binary search (bisection method) - is it worth checking continuity

I am implementing a rather simple matlab code, that gets a function $f$ and 3 real numbers $a,b, \epsilon$ where $\epsilon >0$ is a very small positive number (for instance, no larger than ...
2
votes
1answer
19 views

Uniform continuity problem

Let $f(x)$ be continuous on $[0, \infty)$, $f'(x)$ and $f''(x)$ be continuous on $(0, \infty)$. Which of the following statements are true: I. If $f'(x) > 0$ and $f''(x) < 0$, then f(x) is ...
-1
votes
1answer
53 views

Find $k$ so that $f(x)$ is a continuous function [closed]

Find $k$ so that $f(x)$ is a continuous function. $$f(x)=\left\{\begin{array}{ll}x^2 &x\leq2\\ k-x^2 & x>2 \end{array}\right.$$ Does anyone know how to go about this problem? Thanks in ...
1
vote
3answers
77 views

How to prove, that $e^x$ is uniformly continuous if $x$ is negative?

How can one show with only elementary mathematics, that $e^x$ is a uniformly continuous function on $(-\infty;0]$ I started with $\mid e^x-e^y \mid$, knowing that I assume , that $\mid x-y ...
1
vote
1answer
25 views

Derive property from continuity - is this proof valid?

Prove that if $f:R^+ \rightarrow R^+$ is continuous on the positive reals and is decreasing, then for all $a$ there exists an $\eta > 0$ such that $(a-\eta)f(a-\eta) > \frac{1}{2}a*f(a)$. EDIT ...
0
votes
0answers
30 views

A function is discontinuous at all rational points and continuous at all irrational points [duplicate]

Define $f(x)$ for $x\in[0,1]$ by $f(\frac pq)=\frac1q$ if $p$ and $q$ are relatively prime, and $f(x)=0$ if $x$ is irrational. How can we see that $f$ is discontinuous at all rational points and ...
2
votes
1answer
21 views

if $f([a,b])=[c,d]$ and $[c,d] \subset [a,b]$, is there $x \in [c,d]$ such that $f(x)=x$?

I'm trying to prove something that I'm not sure is correct. Let $f$ be a continuous, differentiable and monotonic function $f:[a,b] \to [c,d]$, where $[c,d] \subset [a,b]$. Is there an $x \in [c,d]$ ...
0
votes
1answer
40 views

What am I doing wrong in this continuity check?

I want to show that the function $f$ is discontiunous. $f$ is defined as follows: $$f(x,y) := \begin{cases} (x^2+y^2)\cdot\sin(\frac{1}{\sqrt{x^2+y^2}}) & , (x, y) \neq (0,0) \\ 0 ...
0
votes
1answer
57 views

$f_a(x) = e^{ax}$ is uniformly continuous over $[0, \infty)$?

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ defined by $f_a(x) = e^{ax}$. a) Prove that $f(x) = e^x$ is not uniformly continuous. b) Determine for wich values of $a$ the function $f_a(x)$ is ...
0
votes
0answers
21 views

Let $f:(0,\infty) \to \mathbb{R}$ s.t f'(x)>x. Prove that f is not uniformly continuous [duplicate]

I'm trying to prove the following statement: Let $f:(0,\infty) \to \mathbb{R}$ s.t f'(x)>x. Prove that f is not uniformly continuous. My first step was thinking about Lagrange, so I wrote that ...
1
vote
2answers
49 views

Proving $ f(x)=(\frac {\sin x} {x})^{\frac {1} {x^2} }$ is uniformly continuous on $(0,1]$

Prove that $f(x)=\Large(\frac {\sin x} {x})^{\frac {1} {x^2} }$ is uniformly continuous on $(0,1]$. Basically what I need to show here is that there is a limit 'from the right' for $x=0$ so the ...
1
vote
1answer
109 views

showing $\int _a^b\left(f'\left(x\right)\right)dx\:=\:f\left(b\right)-f\left(a\right)$

Let $f(x):[a,b]\to \mathbb R$, be differentiable on $[a,b]$ (and continuous) so that $f'(x)$ is integrable on $[a,b]$. I need to show that: $$\int _a^b\left(f'\left(x\right)\right)\mathrm dx = ...
0
votes
1answer
58 views

Why this way of showing that $\sin x$ isn't uniformly continuous is wrong?

I know $\sin x$ is uniformly continuous and it was asked before (Prove $\sin x$ is uniformly continuous on $\mathbb R$). My question is related to this answer: ...
1
vote
3answers
56 views

Show $f$ is uniformly continuous

Let $f$ continuous function on $[0,\infty)$. Lets assume there are $a,b$ such that: $\lim_{x\rightarrow \infty} f(x)-(ax+b) = 0$. Prove $f$ is uniformly continuous on $[0,\infty)$. Well, At ...
1
vote
1answer
33 views

uniform continuity on $(a, b]$ implies limit at $a^+$ exists and finite

Let a uniformly continuous function $f$ on $(a, b]$. Prove that $\lim_{x\rightarrow a^+} f(x)$ exists and finite. What I did so far: from the definition of uniform continuity: ...
2
votes
1answer
38 views

Is this function continuous on transcendental number

This question is motivated from Thomae's function continuity at irrationals together with the fact that transcendental numbers are dense in real numbers. Let $$f(x) = \begin{cases}1 &, \text{x ...
1
vote
2answers
29 views

How to show $\{f_n\}_{n=1}^\infty$ has uniformly convergent subsequence on [0,1]?

Let $\{f_n\}_{n=1}^\infty$ a sequence of second order differentiable functions on the interval [0,1]. If $\forall n\in \Bbb N$ $f_n(0)=f_n'(0)=0$ and for all $n\in \Bbb N$ and $x \in [0,1]$ , ...
0
votes
1answer
21 views

Inverse of Continuous Function on Closed Bounded Part of R. Why Bounded?

Consider the following proposition: Let $A$ be a closed bounded part of $\Bbb R$. Assume $f: A\rightarrow \Bbb R$ is a continuous injective function. Then $f^{-1}: f(A) \rightarrow A$ is also ...
1
vote
3answers
44 views

Show $\lim \limits_{x \rightarrow c_{-}} f(x) \neq \lim \limits_{x \rightarrow c_{+}} f(x)$ imply $f$ is discontinuous at $c$

How to show $\lim \limits_{x \rightarrow c_{-}} f(x) \neq \lim \limits_{x \rightarrow c_{+}} f(x)$ imply $f: \mathbb R \rightarrow \mathbb R$ is discontinuous at $c$ ? I know that $f$ cannot have ...
1
vote
2answers
57 views

Find $\alpha$ and $\beta$ so that $f(x)$ is continuously differentiable

The function $f(x)$ is defined as following $$ f(x) := \begin{cases} \cos x+e^x, & \text{if $x < 0$} \\ \ \alpha(1+x)^{2009}+\beta e^{-x}, & \text{if $x \ge 0$} \end{cases} $$ I need to ...
2
votes
2answers
67 views

Why can a discontinuous function not be differentiable?

I don't really understand why a discontinuous function cannot be differentiable. In Stewart's Calculus, the definition of a function $f$ being differentiable at $a$ is that $f'(a)$ exists. Earlier it ...
0
votes
1answer
19 views

Define multiple-variable function to be continuous

Define the function $f(x,y)= {{x^2 + y (x^2 + y)} \over {x^2 + y^2}}$ at $[0,0]$ so that the function would be continuous. I need help with this calculus problem. I mean, I guess it involves some ...
2
votes
1answer
66 views

Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$

Let $f\in C^{\infty} ([-1,1])$ and suppose there's a constant $M>0$ such that $|f^{(j)}(x)|\le M:\forall j\in\mathbb N$ (including zero) and for all $x\in [-1,1]$. Prove that if $f(\frac 1 ...
0
votes
0answers
26 views

Continuity and differentiability of a function. [duplicate]

How to prove that $f(x)$ is discontinuous at $x=0$ ? $$f(x)=\begin{cases} \sin\left(\tfrac1x\right),& \text{when $x\neq0$} \\ 0, & \text{when $x=0$} \end{cases}.$$
0
votes
0answers
27 views

Questions from a calculus assignment about a function [duplicate]

Can anyone guide me through this problem? Let $f(x) = \lvert 4-x^2 \rvert$, $-4\le x\le 1$. Sketch (I have completed this part). Rewrite $f$ as a piecewise function. Give the range of $f(x)$. Use ...
0
votes
1answer
38 views

If a differentiable function has bounded derivative, Must it be that its derivative continuous?

I got this question: Let $f$ be a continuous function on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, If $f'$ is bounded on $(a,b)$, Must it be the case that $f'$ is ...
1
vote
1answer
100 views

How to prove uniform continuity problem!

A) $f(x)=x^3$ , give an example of an interval where $f$ is uniformly continuous and another where it is not. explain your choose of examples B) decide if $f(x)= \dfrac{1}{\sin x} - \dfrac{1}{x}$ is ...
1
vote
0answers
37 views

Proving $\lim_{x\to c} g\circ f(x)= g(b)$ without sequential criterion

Pardon my English beforehand. I want to prove, without using the sequential criterion for continuity, the next theorem: Let $f$,$g$ be defined on $\mathbb R$ and let $c\in \mathbb R$. Suppose ...
1
vote
2answers
32 views

How would I finish this continuity proof?

I have a multivariable function $f$ with $$f(x, y) = \begin{cases} \frac{x^2+y^2}{y} & \text{if }y \neq 0\\ 0 & \text{if }y = 0 \end{cases}$$ and want to show that it is continuous at $(0, ...
1
vote
1answer
38 views

Uniform continuity of $\arctan x$

Check if $\arctan x$ is uniformly continuous on $\mathbb R$ If I'll show that it's contious on $[0,\pi/2]$ then because it's periodic it would be continuous on $\mathbb R$. So by the definition ...
0
votes
1answer
23 views

Equivalence of different definitions of continuity

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a real function. $f$ is continuous at point $c$ iff $$(\forall\epsilon>0)(\exists\delta>0)(\forall ...
2
votes
1answer
24 views

$ \int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}} \sin(t^2)\; dt = \frac{(-1)^n}{c}, \text{ where } \sqrt{n\pi} \leq c \leq \sqrt{(n+1)\pi}. $

The following is a problem from Apostol Vol 1 Calculus from the section: Continuity. Since Differentiation hasn't been introduced yet, the objective is to solve it without direct reference to ...
1
vote
1answer
28 views

Functions continuity

I have a question regarding continuity of a function that has 2 parts and 2 variables: $$f(x) = \begin{cases} \dfrac{\arctan x}{1 + x^2}, & \text{for $x\ge $ 0} \\ A e^x + B, & \text{for ...
8
votes
1answer
247 views

Does there exist a continuously differentiable function with the following properties?

Does there exist a continuously differentiable function $f: [1,5] \rightarrow \mathbb{R}$, such that $f(1) \lt 0, f(5) \gt 3$ and $f'(x) \leq e^{-f(x)}$? Now do I just integrate it to get $f(x) ...
2
votes
1answer
72 views

Is $f(x)=\log(1+x^2)$ uniformly continuous on $(0,\infty)$?

Is $f(x) = \log(1+x^2)$ uniformly continuous on $(0,\infty)$? My work: Looking at the graph and knowing that $\log$ considered a "slow-growing" function, my guess is that $f(x)$ is uniformly ...
2
votes
0answers
42 views

A question about differentiable functions and step/jump discontinuities

I got this question: Let $f$ be a differentiable function defined on an interval $I$, Must it be the case that $f'$ (the derivative of $f$) doesn't have step/ jump discontinuities on the interval $I$ ...
1
vote
2answers
37 views

Problem of continuous function

Define the function $g(x) = x^2\cos\frac1x$ for $x\ne 0$. What should be the value of $g(0)$ if $g(x)$ is a continuous function? Explain your work and justify your answer. Frankly, I have no ...
1
vote
1answer
22 views

Show that: $\exists x \in \mathbb{R}. \left|P(x)\right| = e^x$

Show that: $\exists x \in \mathbb{R}. \left|P(x)\right| = e^x$. where $P(x)$ is a polynomial different from the zero-polynomial. Obviously, for every $y \in (0, \infty)$ there's $x$ such that $e^x ...
0
votes
0answers
22 views

Enlarging the space $PC(a,b)$ to include functions with one or more infinite singularities

I'm reading a Fourier analysis book and on the chapter about convergence and completeness of orthogonal sets of functions I have one part which I don't understand. I have uploaded the part as an image ...