-2
votes
1answer
32 views

Some questions about uniform continuity

I got the following questions: Let $f$ be a real valued function of a real variable: (1) If $f$ is continuous and bounded on the interval (a,b) (meaning there exist $M,L\in \mathbb{R}$ such that ...
0
votes
0answers
16 views

A question about a function that is uniformly continuous and nonzero on an interval

I got this question: Let $f$ be a function defined on an interval $I$ and let $0<L$ be a constant, If $f$ is uniformly continuous on $I$ and $\forall x\in I, L \leq f(x)$, Must it be the case that ...
0
votes
1answer
27 views

A question about continuous functions from a closed interval into itself

I got this question: Let $f, g:[a,b]\to [a,b]$ be functions that are continuous on [a,b] such that $g$ is onto $[a,b]$, Must it be the case that $\exists x \in [a,b]$ such that $f(x) = g(x)$? This ...
0
votes
1answer
23 views

A question about the relation between supremums of two functions that satisfy certain properties

I got this question: Let $f, g : [a,b] \to [a,b]$ be functions that satisfy $\forall x \in [a,b], g(x) < f(x)$. Prove or give a counter example for the following statements: (1) If $f$ and $g$ ...
1
vote
2answers
37 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 ...
3
votes
1answer
44 views

The continuity of a function at a point x0

Choose correct options , more than one may be correct . (1) $$ \textrm{The function defined by } \begin{cases} f(x)=\cos\left(\dfrac{1}{x}\right) & x\neq 0\\ f(0)=0 & \\ \end{cases} ...
0
votes
1answer
19 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 ...
-1
votes
1answer
42 views

Real analysis: Continuity of a function

Define $f: [0,1) \cup [2,3] \rightarrow [0,2]$ by $$f(x)=\begin{cases} x & x \in [0,1) \\ x-1 & x \in [2,3] \end{cases}$$ Is the function continuous at $x=1$? Is the function continuous at ...
1
vote
1answer
29 views

How to prove whether the absolute value?

I would like to prove with epsilon-delta definition whether the following function is continuous: $f:\mathbb{R} \to \mathbb{R}: x \mapsto \vert x \vert$ I tried to begin with $\vert x \vert -\vert a ...
0
votes
1answer
23 views

Taylor expansion with non integer exponents in the rest

Consider the function: $$f(x)=\sqrt[3]{8x^2+4x+1}$$ 1) Find $a,b,\alpha,\beta$ such that: $$f(x)=ax^\alpha+bx^\beta+o(x^{-1/3})$$ 2) Find $A=f([0,+∞[)$ and prove that $f:[0,+∞[\rightarrow A$ is ...
3
votes
1answer
36 views

Proving continuity of polynomial

How can I prove that a polynomial with degree $n$ is continuous everywhere in $\mathbb{R}$ using definitions? With induction. I can show that this polynomial is continuous at $x_0$ but I do not know ...
1
vote
1answer
34 views

Continuity of floor function

Am I right in saying that $f(0)=0$ for $f(x)=1/\left(\lfloor 1/x\rfloor\right)$. Is it also true that it is continuous at 0 and $x$'s near $0$?
0
votes
1answer
49 views

Spivak problem on property of continuous functions.

Ok so problem goes like this: If f is continuous on [0,1] and f(x) is in [0,1] for each x.Prove that f(x)=x for some x. My proof goes like this but I am not quite sure of my result. Let ...
1
vote
2answers
42 views

What is the difference between “differentiable” and “continuous”

I have always treated them as the same thing. But recently, some people have told me that the two terms are different. So now I am wondering, What is the difference between "differentiable" and ...
1
vote
0answers
109 views

Continuity of a parametric integral (where the integrated function is discontinuous)

For all $t\in\mathbb{R}$ consider $$F(t):=\int_\mathbb{R}e^{-x^2/2}\log|t+e^x|\,dx \;.$$ I managed to show that $F(t)$ is well-defined and finite for every $t$. I would like to show that $F$ is ...
0
votes
1answer
47 views

If the sum of two functions is continuous at a point, can one function be continuous and the other not

I got this question: Let $f$ and $g$ be functions such that $f+g$ is continuous at $a$, Must it be the case that both $f$ and $g$ are continuous at $a$ or that both $f$ and $g$ are discontinuous at ...
0
votes
2answers
65 views

Prove that $e^{x}-2\cos(x) = 0$ where $x\in(0,1)$ has solution.

Prove that $e^{x}-2\cos(x) = 0$ where $x\in(0,1)$ has solution for $x$. I'd like to do this without derivatives, just using limit definition and function continuity. To begin, we could rewrite this ...
1
vote
1answer
62 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)| < ...
0
votes
3answers
46 views

Is function continuous $x\sin(y)/(x^2+y^2)$

I have the following function and I can't seem to prove that it is not continuous: $ f(x,y) = \begin{cases} 0, & {(x,y) = (0,0)} \\ x\sin(y)/(x^2+y^2), & \text{else} \\ \end{cases}$
-1
votes
1answer
40 views

continuity and differentiability of function of two variables

Let $f(x,y)$ be $$f(x,y): \begin{cases} x & \text{for } y = 0\\ x-y^3\sin\left(\frac{1}{y}\right)& \text{for } y \neq 0\end{cases} $$ then check continuity and differentiability at $(0,0)$. ...
1
vote
2answers
42 views

What kind of functions can be Riemann integrable?

I have learned that every continuous, or piecewise continuous function can be Riemann integrated. But then, are there uncontinuous functions that are Riemann integrable? And if there is, can I still ...
1
vote
1answer
47 views

Showing if $f_n \to f$ uniformly and each $f_n$ has at most $10$ discontinuities, then so does $f$

Suppose that $f_n:[a,b] \to \Bbb R$ and $f_n$ uniformly converges to $f$ as $n$ goes to infinity. How to prove that if each $f_n$ has at most ten discontinuities (the discontinuities for each $f_n$ ...
3
votes
2answers
62 views

A function vanishing at infinity is uniformly continuous

If $f\in C_0(\mathbb{R})$ (i.e. $f$ continuous and for all $\varepsilon>0$ there is $R>0$ such that $|f(x)|<\varepsilon$ whenever $|x|>R$), then why is $f$ uniformly continuous? I know ...
0
votes
0answers
47 views

Continuity of a function that maps non convergent sequences onto non convergent sequences

Let be f a surjective real function defined on R mapping every non convergent sequence onto non convergent sequence. Prove that f is continuous. I have proved that f is injective. Can you help me ...
1
vote
1answer
40 views

Continuity of $\sqrt[\leftroot{0}\uproot{0}3]{\sin(\theta)}$ on interval $[-\pi, \pi]$

I'm doing a problem where my task is to determine whether a given function is continuous, piecewise continuous or piecewise smooth on interval $[-\pi, \pi]$. According to my book the function: ...
1
vote
2answers
82 views

$f(x,y)=x^y$ is not continuous at (0,0)

How to prove that the function $f(x,y)=x^y$ ($x>0$) is not continuous at the point $(0,0)$? I tried $y=x^\alpha$ $(\alpha>0)$ but this does not work since for such $y$ we have $x^y \to 1$ as ...
9
votes
1answer
315 views

Proving the existence of a point with a certain property for a continuous function

Let $f:[0,1]\to\mathbb{R}$ a continuous function and $\int_0^1xf(x)dx=0$. Show that there exists a point $c\in(0,1)$ so that $f(c)=(\int_c^1f(x)dx)^2$. As a potential solution, I tried assuming that ...
1
vote
1answer
42 views

How can a graph have a jump discontinuty at $x \not= 0$ and removable discontinuity at $x = 0$?

I stumbled an a past test question where the student was asked to provide any example of a graph that has those two particular properties. I now there should be a piece-wise function, an maybe a case ...
1
vote
1answer
37 views

Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$

$f(x)$ is continuous on $[a,b]$. Now we define a new function $M(t)$, for every $t\in[a,b]$ $$M(t) = \sup_{a \leq x \leq t} f(x).$$ Prove formally that $M(t)$ is continuous on $[a,b]$. (sup = ...
0
votes
1answer
23 views

Differentiability of trigonometric piecewise functions

So I have a function of a real variable $x$: $f(x) = \left\{\begin{array}{lr} x \int_0^{tanx} \dfrac{t^2}{\sqrt{1+t^3}}dt & if \: x \ge 0\\ sin^2(x) & if \: x \lt 0 ...
1
vote
2answers
84 views

Showing $f(x)\le 2$ for all $x\in [0,1]$

Let $F:[0,1]\to \mathbb R$ be continous such that for all $x\in \mathbb Q \cap [0,1]$ we have $f(x)\le 2$. Show that for all $x\in [0,1]$ we have $f(x)\le 2$. I'm going to write the answer I got ...
1
vote
0answers
46 views

How to prove a function is continuous?

What is a general outline of a proof of the continuity of a function? My background is Calculus 2. I think it uses the $\epsilon$-$\delta$ definition of a limit, but I only have a very vague idea of ...
0
votes
1answer
47 views

Can such a function be continuous? [duplicate]

Let $f$ be a function from $\Bbb R$ to $\Bbb R$ such that $f(x)$ is rational when $x$ is irrational, and $f(x)$ is irrational when $x$ is rational. Can $f$ be continuous? Thanks for your help.
1
vote
1answer
45 views

Real analysis homework problem

Let $h:[0,1] \times [0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Assume that there is a constant $0<c<1$ such that $|h(x,y,s)-h(s,y,t)| \leq c|s-t|$ for all $x,y \in [0,1]$ ...
0
votes
2answers
94 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 ...
5
votes
4answers
78 views

Continuous function on $\mathbb{Q}$

Let $f:\mathbb{Q}\to\mathbb{R}$ be a function defined as: $$f(x) = \begin{cases} 0 & x^2 < 2\\ 1 & x^2 \geq 2 \end{cases} $$ Is this function continuous? How can we check the ...
3
votes
1answer
269 views

If $f$ is a twice differentiable function and $f(2^{-n}) = 0 $, for all $n \in \mathbb N$, then $f^\prime(0) = f^{\prime\prime}(0) = 0$.

Let $f : \mathbb R \to \mathbb R$ be a twice differentiable function, such that $f(2^{-n}) = 0$, for all $n \in \mathbb N$ . Show that $$f^\prime(0) = f^{\prime\prime}(0) = 0.$$ My attempt. First, ...
0
votes
2answers
43 views

Proving a property of piecewise continuous functions

How to prove the following problem: Suppose $f \in PC(a,b)$, where $PC(a,b)$ means the set of piecewise continuous functions on the interval $[a,b]$ and $f(x) = \frac{1}{2}[f(x-) +f(x+)]$ for all $x ...
1
vote
2answers
50 views

How can I prove continuity of this function?

I have to prove that the function $f:\Bbb{R}\times\Bbb{R}\to\Bbb{R}$ defined as follows: $$f(x, y)=\frac{xy}{x^2+y^2}\text{for } (x, y)\neq (0,0)$$ $$f(x, y)=0\text{ for }(x, y)=(0,0)$$when taken as a ...
0
votes
0answers
52 views

calculus continuity question.please help.

Show that $\sin (x + y)$ and $\cos(x-y)$ are continuous at $(0,\pi/2)$ using $\epsilon$ and $\delta$ definition. I have tried it as follows. Let $\epsilon>0$ be any real number. To find $\delta ...
18
votes
1answer
253 views

$f(f(\sqrt{2}))=\sqrt{2}$ then f has a fixed point

$f(x)$ is continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ $f(f(\sqrt{2}))=\sqrt{2}$ Prove that $f$ has a fixed point in other words prove the there is $x_1$ such that $f(x_1)=x_1$ I tried using ...
4
votes
1answer
31 views

Continuous function and its extrema

Let $f(x)$ be a continuous function in $(a,b)$ and it has in this interval $m$ local maximum points and $n$ local minimum points. Then : $|m-n|\leq1$ It seems very obvious but is there any simple ...
1
vote
1answer
84 views

Show that the function $ f(x)=\frac{1}{x+1}$ is continuous at $x=1$ using the $\varepsilon$-$\delta$ definition of continuity

Show that the function $ f(x)=\frac{1}{x+1}$ is continuous at $x=1$ using the $\varepsilon$-$\delta$ definition of continuity. My initial thoughts were that this is a straight plug in of value at ...
0
votes
1answer
59 views

Monotone convergence of continuous functions

Is it true that for a sequence of functions $f_n \in C(K,\mathbb{R})$, where $K \subset \mathbb{R}^n$ the limit is semicontinuous? I would say that if it's a monotone decreasing lsequence, then the ...
2
votes
1answer
41 views

Continuosly differentation on composite functions

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ a $C^1$ function and defined $g(x) = f(\|x\|)$. Prove $g$ is $C^1$ on $\mathbb{R}^n\setminus\{0\}$. Give an example of $f$ such that $g$ is $C^1$ at the origin ...
0
votes
1answer
66 views

calculus continuity of a hard question?

How do i calculate the continuity of a function if the functions are in a given set limit. I tried doing it but I epically failed… Please help!how do I solve it? When I tried to solve this I got that ...
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
2answers
30 views

Is the function continuous at the indicated point?

$f(x) = x[x]$ at $x=2$ ($[x]$ is the greatest integer function) I am little confused, it seems like the function does exist at the given point. when limit goes to 2, $2[2] = 4$ and $f(2) = 4$ so ...
0
votes
1answer
23 views

If the function $f(x) =\lfloor \frac{(x-2)^3}{a}\rfloor \sin(x-2) +a\cos(x-2), \lfloor . \rfloor$ denotes …

Problem : If the function $$f(x) =\left\lfloor \frac{(x-2)^3}{a} \right\rfloor \sin(x-2) + a \cos(x-2),$$ where $\lfloor . \rfloor$ denotes the greatest integer function is continuous and ...
7
votes
9answers
340 views

Nonpiecewise Function Defined at a Point but Not Continuous There

I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, ...