-2
votes
3answers
59 views

problem on continuity [on hold]

For $x>0$, let $[x]$ denote the largest integer less than or equal to $x$. Let $f:[0,\infty)\rightarrow\mathbb{R}$ be given by $f(x)=[x^2+[x^2]]\sin(2\pi x)$. Then $f$ is continuous at $2$ or ...
4
votes
4answers
135 views

Real Analysis: Showing $f: \Bbb Q \to \Bbb Q$ is continuous

The following is all working in $\mathbb{Q}$, not $\mathbb{R}$. I am working with the function $f: \mathbb{Q} \to \mathbb{Q}$ defined piece-wise by $f(x)=-1$ if $x^2<2$ $f(x)=1$ if otherwise I ...
3
votes
1answer
27 views

Show that $Y$ is not path-connected

Let $\mathbb{R}^2$ with the usual topology and let $$ Y = A_0 \cup (\bigcup_{n \in \mathbb{N}} A_n) \cup (\bigcup_{n \in \mathbb{N}}L_n)$$ where $$ A_0 = \{ 0 \} \times [0,1] \qquad A_n = \{ ...
0
votes
2answers
39 views

Question about limit and continuity

I have that $u_0>0$ , $u_n=u_n^+-u_n^{\raise{1pt}{-}}$ and $u\mapsto u^{±}$ is continuous if $u_n\rightarrow u_0$ why we have that $u_n^+\rightarrow u_0$ and $u_n^{\raise{1pt}{-}}\rightarrow 0 $ ...
0
votes
1answer
32 views

Continuity proof of two-variable function.

The Assignment Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} ...
0
votes
1answer
38 views

Determine if the following function is continuous in $(0,0)$.

Assignment: Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} 1& ,x≤ 0, y ...
15
votes
3answers
292 views
+50

Continuity of a function in two variables

Function $f(x,y)$ is continuous in each variable separately. Prove that there exists a point where it is continuous in two variables. I do not quite understand how to act here. I know the ...
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 ...
1
vote
1answer
49 views

Show that $f*(x) = \sup \{ f(y) : a \leq y \leq x \}$ is a non-decreasing continuous function

I am currently working on a problem and stuck on it. Here is the problem (it comes form Elementary analysis, the theory of Calculus by K. Ross P.153): Q: Let $f$ be a continuous function on [a,b]. ...
2
votes
0answers
41 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 $I$ ...
0
votes
0answers
26 views

how to find the coefficient for a function to be continous at all $x$

I'm having a problem solving this question, we have just learnt it at school today and this is my homework. Could you help me please? Find the values of a such that $f$ is continous for all values of ...
-1
votes
2answers
35 views

Is it continuous at $(0,0)$?

$$f(x,y)=\begin{cases} \frac{xy}{x^2+y^2}, \text{ if } x^2+y^2\neq 0 \\ 0, \text{ if } x^2+y^2=0 \end{cases}$$ Is it continuous at $(0,0)$?
1
vote
2answers
80 views

How to prove that a function is continuous?

Could you give me some hint how to solve this question: Suppose $f$ is a differentiable function for all $0<x<1$,$f(0)=1,f'(x)>0$ in the given interval. It is obvious that $f$ is continuous ...
0
votes
2answers
51 views

Definition of continuity question

Hey sorry about the picture, its of an example in a lecture slide. Just a quick question is I understand how they get the answer (in red) if they are told the constraint that delta must be less ...
1
vote
0answers
25 views

Check my answer - show a function is integrable and find the integral

Let $Q =[0,1]$x$[0,1]$. Let $f: Q \to \mathbb R$ defined as such: if $(x,y) \in \mathbb Q$x$\mathbb Q$ then $f(x,y)=\frac{1}{n_1}+\frac{1}{n_2}$ where $x=\frac{m_1}{n_1}$ and $y=\frac{m_2}{n_2}$ are ...
2
votes
1answer
46 views

How to prove that a Lipschitz function is absolutely continuous?

$f:[a,b] \rightarrow \mathbb{R}$ is a Lipschitz function. How to prove that it is absolutely continuous on $[a,b]$? My attempt: Let $\epsilon> 0$. Set $d = \epsilon/M$. If $P = \{[x_i, y_i]\}$ is ...
2
votes
4answers
172 views

Prove the function $e^{-x^2}$ is uniformly contiuous on $[0,\infty)$

I have no idea how to prove this. But my teacher gave me a hint of taking cases of $\epsilon\ge 1$ and $0\lt\epsilon\lt1$ . Please give some insight and thank you so much. Additional information ...
0
votes
1answer
30 views

Prove that f is 1 time continuously differentiable and express f' in terms of f

Suppose that $f$ is a continuous function on R that satisfies $$f(x) = 5 + 2\int_0^x f(t) \,dt$$ Prove that $f \in C{^1}(\mathbb{R})$ [ aka $f$ is one-time continuously differentiable on $\mathbb{R}$ ...
1
vote
2answers
77 views

Show the function is integrable and find the integral - somewhat complex question

We are given $Q = [0,1]$x$[0,1]$ We are also given the function $f(x,y) = (\frac{1}{10})^n$ where $\frac{1}{2^{n+1}} < \max(x,y) \leq \frac{1}{2^n}, (n=0,1,2,...)$ and $f(0,0)=0$. Show that $f$ ...
1
vote
1answer
41 views

A question about a continuous function that satisfies the property $\forall x\in\mathbb{R},\exists x<y\in\mathbb{R},f(x)<f(y)$

I got this question: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function that satisfies the property: forall $x\in\mathbb{R}$ there exists $y \in\mathbb{R}$ such that $x < y$ and ...
1
vote
1answer
36 views

A question about a continuous function that satisfy certain limits at $\pm\infty$

I got this question: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $\lim_{x\to\infty}\frac{f(x)}{x^2}$ and $\lim_{x\to -\infty}\frac{f(x)}{x^2}$ exist and are real numbers. ...
1
vote
1answer
33 views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ that got no extrema must be one to one

I got this question: Prove that if $f:\mathbb{R}\to\mathbb{R}$ is a continuous function that got no extrema then $f$ is one to one. I tried to prove it but I don't know how to proceed. I started by ...
0
votes
2answers
49 views

Intermediate value theorem problem

Problem: The equation $x=-5\cos(x)$ has at least $3$ distinction solutions. Use the intermediate value theorem to show that this is true. I drew the function,but I don't know what to do next.
0
votes
1answer
47 views

Help, check the uniform continuity

(1) $f(x)=sin(1/x)$ on $(0,1]$ ? ( I know it is not uniform continuous on $(0,1)$) (2) $f(x)= xsin(1/x)$ on $(0,1]$? (3) $f(x)=sin(x^2)$ on $[0, \infty)$?
1
vote
2answers
45 views

Limit and integral properties of a continuous function

Let $f$ be a continuous function on $[0,\infty)$ such that $\displaystyle\lim_{x \to \infty}f(x)= c$. Show that $\displaystyle\lim_{x \to \infty} \frac{1}{x}\int_0^x f(s)\;ds = c$. I've tried ...
0
votes
1answer
41 views

A question about continuous function on a closed interval and the supremum

I got this question: Let $f$ be function that is continuous on the interval $[a,b]$ and let $A=\{x \in [a,b] | f(x) = f(a)\}$. (1) Prove that A is a non empty set. (2) Prove that A is bounded above ...
0
votes
1answer
42 views

Proving definition of limits with definition of continuity and visa versa

That is: Let $f: D \rightarrow \mathbb{R}$. Suppose $x_0 \in D$ is a limit point. Prove $f$ is continuous if and only if $\lim_{x\to x_0} f(x) = f(x_0)$. Also, if $x_0$ is not a limit point, prove ...
1
vote
2answers
57 views

$\varepsilon$-$\delta$-definition for continuity of $x^n$

Show that $f:\Bbb R\to\Bbb R,x\mapsto x^n$ with $n\in\Bbb N$ is continuous in $x_0=0$ using the $\varepsilon$-$\delta$-definition. We assume that $$\forall ...
1
vote
3answers
68 views

If Limit of function and derivative exist, then limit of derivative is 0 [duplicate]

Any hints for this question , My attempt; Say $f(x):0$$\rightarrow$$\mathbb{R}$ The by MVT, there exists a $c$$\in$$(0,\infty)$ , such that; $f'(c)=$$\frac{f(x)-f(0)}{x-0}$ but im not sure about this ...
0
votes
1answer
56 views

Differentiability conditions for a piecewise function

So this is an analysis class, and we just started the unit on differentiability -- however I missed the class. Can someone start me off with a good real analysis definition for differentiability of ...
0
votes
2answers
96 views

Discontinuity of the characteristic function

Let $A \subseteq \mathbb{R}^n$. Let $f(x) = \chi_A $ be the characteristic function, and put $D = \{ x : f(x) \; \; \text{is discontinuous} \} $. Then $\partial A = D $. MY try: Let $y \in D $. ...
0
votes
2answers
42 views

Proving a if a property holds for a dense set then it holds on the field that the set is a subset of.

I am currently studying for my analysis exam and have come across this question, I can't seem to grasp the idea of a "dense set" especially with the definition given in the question. When I read it, ...
1
vote
1answer
86 views

Proof that $f(x)=x^{1/n}$ is continuous.

Here's what I've done: According to the definition, a function is continuous at $c$ if, for any $\epsilon>0$, there exists a $\delta>0$ so that, if $|x-c| < \delta$, then $|f(x)-f(c)| < ...
1
vote
0answers
73 views

Proving a function is continuous using preimages

I want to prove that f is continuous using the preimages of open subsets here. Never worked with pre images before -- can anyone help? (also would love a good definition of a preimage).
0
votes
3answers
75 views

Proving or disproving uniform continuity of various real functions

Okay before anyone does anything, I don't want any of you guys to just write out proofs for all of these, that's asking a bit much :P Maybe just do one, and make it detailed because I really need ...
2
votes
1answer
65 views

Question regarding $\epsilon-\delta-$proof

I want to prove the continuity of $f(x) = x^2$. Lets take $\epsilon > 0$ and $|x-x_0| <\delta$. I do: $$|f(x) - f(x_0) |= |x^2 - x_0^2| = |(x-x_0)(x+x_0)| < \delta |x-x_0|$$ Now the ...
1
vote
3answers
93 views

Uniform continuity of a function and cauchy sequences

So I'm pretty sure this is almost immediate from the definitions, please tell me if I am incorrect.. Consider two cauchy sequences in D, $\{x_n\}$ and $\{y_m\}$. Since $f$ is uniform continuous we ...
1
vote
1answer
43 views

Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
1
vote
2answers
48 views

Proof that a complex function is continuous at $z=0$

Given the function $f\colon \mathbb{C}\to\mathbb{C}$ by $f(z)=\begin{cases} \frac{xy(x+iy)}{x^2+y^2} & \text{if } z\neq 0\\ 0 & \text{if } z=0 \end{cases}$ with $z=x+iy$. How do I ...
0
votes
1answer
28 views

Understanding why an IVP has a solution, using uniqueness and existence theory

Given the existence and uniqueness theory: If $f$ is Lipschitz continuous over some region $D$, then there is a unique solution to the initial value problem (IVP): $u'(t) = f(u,t), \hspace{5mm} ...
3
votes
2answers
42 views

Determine best possible Lipschitz constant

I'm slightly confused by a homework problem here...I've been given the function: $ f(u) = log(u) $ With the bounds: $ 2 \leq u \lt \infty $ Now I thought I understood what the Lipschitz Condition ...
0
votes
2answers
129 views

Prove that the Rational function $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ is uniformly continuous

I need some help with a calculus homework question. Here is said question: Let there be two polynomials $q$ and $p$ such that $\deg(p)\leq\deg(q)+1$ and $q(x)\neq0$ for all $x\in\mathbb{R}$. Show ...
1
vote
3answers
143 views

let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume $f(z) = f(2z)$, prove that f is constant

$f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume that $f(z) = f(2z)$ for all $z \in \mathbb{C}$. Prove that f is constant... Then we are supposed to use this result to ...
0
votes
1answer
31 views

Proving Continuity in Multiple Variables

The Exercise: $f(x,y)=xy/(x^2+y^2)$ if $x \ne 0$ $f(x,y)=0$ otherwise Where is $f$ continuous? My Attempt: At $(0,0)$, let y=kx. $lim_{(x,y)\to (0,0)}f(x,y) = lim_{x\to 0}f(x,kx)=k/(1+k^2)$ which ...
2
votes
1answer
43 views

Weak Lower Semicontinuity Generalized to any $L_{p}$ space

I am having difficulty with the following proof: Generalize the weak lower semi-continuity of$L^{p}$ norms to all $1\leq p < \infty$; i.e., show that if $u_{n}\to u$ weakly in $L^{p}$, then ...
1
vote
3answers
44 views

Proving a fact about continuous function

Prove that if $f(a)>0$ and $f$ is continuous, then there is a $\delta >0$ such that for all $x$, $|x-a|< \delta$ implies $f(x)>0$.
1
vote
1answer
49 views

Transformation of a continious function

Suppose that $f:[0,2\pi$] $\rightarrow \mathbb{R}$ is continuous and $f(0)=f(2\pi)$. Show that there exists an $x\in[0,\pi$] such that $f(x)=f(x+\pi)$. I simply have no idea where to start, any help ...
3
votes
3answers
86 views

Prove that there is an $\varepsilon$ such that $f(x) > x + \varepsilon$ for a continuous $f(x) > x$ at $[0,1]$

I know this question was answered by using another theorem here but I wish I could get comments on my way of trying to prove it. We were asked to prove that for a function $f(x) > x $ which is ...
0
votes
1answer
52 views

Equivalence relation between measures $\nu$, $\mu$ is equivalent to $\nu = f \mu$ for a density $f$.

I'm working on an exercise that wants me to show that for $\sigma$-finite measures $\nu$ and $\mu$ the relation $\nu \sim \mu$ (defined by $\nu \ll \mu$ and $\mu \ll \nu$) is equivalent to $\nu = f ...
1
vote
1answer
77 views

Proving discontinuity

Assume set $A$ is countable and let$$f(x)=\cases{1 \text{ if }x\in A\\0\text{ if }x\notin A }.$$ Prove that $f$ is not continuous at $c\in A$. I've seen such a problem before where ...