-3
votes
2answers
57 views

Continuity of $f$ define by $f(x,y)=\frac{x^2+y^2}{\tan(xy)}$? [closed]

consider the function $f$ define by $$f(x,y)=\begin{cases}\frac{x^2+y^2}{\tan(xy)}&\text{if}\ (x,y) \neq (0,0)\\ 0&\text{if}\ (x,y) = (0,0) \end{cases}$$ Prove that the function is ...
1
vote
0answers
23 views

Lipschitz continuity power type function [duplicate]

Is the function $f(x)=x^{\gamma+1}$, where $x>0 $ and $\gamma<0$ Lipschitz continuous ? I am a bit confused !
3
votes
4answers
187 views

Prove/disprove: if $\lim\limits_{ n\to\infty} f(n)=\infty$ then $\lim\limits_{ n\to\infty}f(f(n))=\infty$

Let $f(x)$ a continuous function on $\Bbb{R}$. Prove/disprove: If $\lim\limits_{n\to\infty} f(n)=\infty$, then $\lim\limits_{n\to\infty}f(f(n))=\infty,$ where the limits are taken over $n \in ...
1
vote
1answer
41 views

What happen to composite of infinite number of continuous functions?

We all know that a composite of continuous functions is continuous. And this holds for any $\textbf{finite}$ number of functions. My question is what happen to infinite number of functions? Is it ...
1
vote
1answer
36 views

Is a function with a random variable continuous?

I often like to fool around on graphing calculators when I am bored. A function that can be very amusing is $f(x) = rand \times sin x$ Now, on my TI-84 Plus, this looks obviously discontinuous ...
2
votes
1answer
69 views

If a function is both upper and lower semicontinuous, does it have to be continuous?

I am looking for an example of a function which is both upper and lower semi continuous but is not continuous. I have an example: $$f(x):=\begin{cases} 1 & \mathrm{if}\; x < 1,\\[7pt] ...
0
votes
1answer
38 views

how to find uniform continuity

I have some questions on continuity. What is the difference between continuous and uniformly continuous function? Please explain with this question. Find $f(x)=x^2$ is uniformly continous on ...
1
vote
0answers
30 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 ...
1
vote
2answers
114 views

Epsilon-Delta continuity definition for straight lines parallel to axes

I am taking a course on real analysis online and I encountered the $\epsilon-\delta$ definition for a function to be continuous. But I wonder if I can apply it to functions which are straight lines ...
1
vote
1answer
54 views

$C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
-2
votes
3answers
64 views

problem on continuity [closed]

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 ...
0
votes
3answers
59 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
0answers
39 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
2answers
43 views

showing the function is continuous at a point using $\epsilon$ and $\delta$

I have this question: Use the definition of continuous function with $\epsilon$ and $\delta$ to show that the function $f$, defined as $$f(x)=\begin{cases}0&\textrm{if } x=0 \\x \sin\frac{1}x ...
-1
votes
1answer
55 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
1answer
26 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
3answers
63 views

If $f$ is continuous, so is $g=|f|$ [closed]

Prove that if $f$ is continuous, so is $g=|f|$. I need help on this. Thank you. Ok, this is my first time here. The definition of continuity i am using is that $f$ is continuous at $a$ if for any ...
1
vote
3answers
63 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 ...
0
votes
0answers
45 views

implicitly define a function

The first part i made $u=\frac{z}{x}$ and $v=\frac{y}{x}$ and after calculating the partial derivatives $\frac{dz}{dx}$ and $\frac{dz}{dy}$ The second i have no idea how to do it
2
votes
3answers
65 views

prove that $f(x)=\sum _{n=0}^{\infty}\frac{\cos(nx)}{2^n}$ is continuous

I refered that each fn is continuous because its the fraction of a continuous function by a number and so $f(x)$ that is the sum of continuous functions is continuous. Is it right?
0
votes
2answers
32 views

Combination of continuous and discontinuous functions

I know that combining two continuous functions gives a continuous function, i.e., if $f(x)$ and $g(x)$ are continuous, then $f(x)\pm g(x)$, $f(x)\times g(x)$ and $f(x)\div g(x)$ are continuous ...
2
votes
2answers
69 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 ...
9
votes
5answers
952 views

Functions that are continuous only at two points?

I need to find a function $f:\mathbb{R}\to\mathbb{R}$ which is continuous only at two points, but discontinuous everywhere else. How on earth would I go about doing this? I can't think of any ...
0
votes
1answer
23 views

help with continuous and differentiable theorems

consider $b>a>0$ and $f:\left[a,b\right]\rightarrow \mathbb{R}$. $f$ is continuous at $\left[a,b\right]$ and differentiable at $\left(a,b\right)$. also, ...
8
votes
1answer
52 views

About continuity

one question is disturbing me : let f and g two continuous (real-valued) functions on the unit interval [0,1] with the property that $[f(x)-f(y)][g(x)-g(y)]=0 ,\forall x,y \in [0,1]$. To my ...
1
vote
2answers
20 views

Investigating a function with a parameter

I got stuck on solving this problem: For which $a \in \Bbb R$ is the function $$ f_a: \ ]1, \ \infty[ \; \longrightarrow \ \Bbb R: x\mapsto \frac{\log x}{(x-1)^a} $$ continuous on $[1, \ ...
3
votes
1answer
26 views

Show for whih values this following function is continuous

For the function $f: [0,2 \pi] \rightarrow \mathbb{R}$ ,state at which points $c \in [0, \pi]$ is $f$ continuous or discontinuous. $$f(x)=\begin{array}{cc} ( & \begin{array}{cc} ...
2
votes
1answer
45 views

How to prove that $f_n(x) = \frac{1}{1+n^{2}x^{2}}$ is continuous on $[0,1]$?

I am having trouble verifying continuity. This seems like a very simple problem but I am not sure if my approach is correct: To prove that $f_n(x) = \frac{1}{1+n^{2}x^{2}}$ is continuous on $[0,1]$, ...
2
votes
2answers
68 views

Constructing a function similar to x^3 between [0,1]

I'm trying to construct a function $f$, in order to normalize a dataset(obviously where all the element come from $[0,1] \in \mathbb{R}$. The big picture is that the envisioned $f: [0,1] \rightarrow ...
0
votes
1answer
95 views

A continuous function on the real line such that the preimage of every point is either empty of has exactly 3 points [duplicate]

Let $f : \mathbb{R} \to \mathbb{R}$ be a function with all fibres $(\lbrace{x \in \mathbb{R}| f(x) = c\rbrace} = f^{−1}(c)$, $c \in \mathbb{R})$ either empty or consisting of exactly three points. ...
1
vote
1answer
37 views

Replacing both f(a) and f(b) with f(c) (confused, please help)

I came across a step in a proof which is puzzling me. The step basically makes the following claim: If $f$ is continuous on some interval $I$, and $a,b \in I$, then there exists $c \in I$ such that ...
2
votes
1answer
70 views

Monotonic function satisfying darboux property $\Rightarrow$ continuous

Assume $f : I \rightarrow \mathbb{R}$ is a non-decreasing on an open interval $I$ and that $f$ satisfies the Intermediate value property or Darboux's property on $I$ (that is, for any $a < b$ ...
4
votes
3answers
88 views

If $f$ is continuous then there exists $x\in [0,1]: f(x)=x$

I wish to prove the following by contradiction: Let $f:[0,1]\rightarrow[0,1]$ be a continuous function. Prove that there exists $x\in [0,1]$ such that $f(x)=x$. Proving this directly, one would ...
-1
votes
1answer
30 views

Continuity of a Function

I'm dealing right now with properties of a function and I have to prove if a given function is injective, surjective or bijective. I prove injectivity with the formula $x_1 = x_2 \Rightarrow f(x_1) = ...
6
votes
1answer
266 views

Can monsters of real analysis be tamed in this way?

Consider the Weierstrass Function (somewhat generalized for arbitrary wavelengths $\,\lambda > 0$ ): $$ W(x) = \sum_{n=1}^\infty \frac{\sin\left(n^2\,2\pi/\lambda\,x\right)}{n^2} $$ $W(x)$ is an ...
2
votes
2answers
95 views

Prove that f is differentiable at $0$! Not continuous though, Right!?

Suppose $f(x)$ equals $x^2$ when $x\in \mathbb{Q}$ and $0$ when $x \not\in \mathbb{Q}$ Prove that $f$ is differentiable at $0$ and find the derivative $f'(0)$ Shouldn't this be obvious, since $x^2$ ...
2
votes
1answer
74 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 ...
1
vote
2answers
223 views

Find a function that satisfies the condition.

Let $\epsilon > 0$ be fixed and $t$ a variable that takes values in the universal covering space of ${\mathbb{C} \setminus \{0\}}$. Find a continuous function $f(s$) such that $$|t \log t| = |t| ...
1
vote
1answer
31 views

Can someone explain the concept of continuity and differentiability for functions of several variables?

Can someone explain the concept of continuity and differentiability for functions of several variables? Illustrated examples will definitely help, on how to solve problems(or establish proofs) of the ...
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 ...
-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
82 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 ...
1
vote
0answers
29 views

Suppose limit as x approaches 0 of the derivative of f(x)=a. Show that the derivative at 0 exists and it is equal to a. [duplicate]

Let f be a continuous function on the interval [0,1], which is differentiable on (0,1). I know that I will have to use the definition of a derivative, which is the limit as x approaches 0 ...
4
votes
3answers
66 views

Find a continuous function on the reals where $f(x) >0$ and $f'(x) < 0$ and $f''(x) < 0$

We need to find a function $f(x)$ where $f(x) >0 $and $f'(x) < 0$ and $f''(x) < 0$ where $f$ is continuous for all real numbers. We have tried $ f(x) = \sqrt{-x}$ however this is not defined ...
2
votes
1answer
21 views

Help with choosing delta-epsilons regarding continuity

Is there any good strategy for picking an epsilon-delta in local and global continuity? I'm really struggling to pick such values. For example, I'm not sure entirely how the author got the values ...
0
votes
1answer
60 views

How to find if and where $f(x)$ is continuous and/or differentiable for a given piecewise function? [closed]

What approach would be ideal in finding if and where $f(x)$ is continuous and/or differentiable when $f$ is a piecewise defined function? A concrete example is below, but I'm interested in general ...
1
vote
0answers
33 views

A bijective function $f$ between two compact Hausdorff spaces is continuous if $f$ preserves compact sets [duplicate]

I am trying to prove that if $f: X \longrightarrow Y$ is a bijection between two compact Hausdorff spaces such that $f[W]$ is compact in $Y$ for all compact $W$ in $X$, then $f$ is continuous. Here ...
0
votes
4answers
72 views

A continuous surjective function from $(0,1]$ onto $[0,1]$

I'm trying to construct a continuous surjection from $(0,1]$ onto $[0,1]$, but I'm not getting anywhere. I don't immediately see a contradiction which falsifies the existence of such a function, so my ...
5
votes
1answer
99 views

Is the standard part function another devil's staircase?

The devil's staircase or Cantor function is an awesome function that increases value but has derivative zero everywhere (or "almost", whatever that means). I was incredibly amazed when I found out ...
1
vote
1answer
42 views

Conditions yielding a unique fixed point of a continuous differentiable function.

Let $f$ be a function defined on $[0,1]$ which is continuous for each point in $[0,1]$ and differentiable for each point in $(0,1)$. Suppose that $f^\prime (x) \neq 1$ for every $x \in (0,1)$. ...