Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, integration through the fundamental theorem of calculus.

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3
votes
3answers
59 views

$\lim_{x \to 0} \frac{f(x)-c}{g(x)}$ for $c \in \mathbb{R}$

My question involves evaluating the limit of the function: $\lim_{x \to 0} \frac{f(x)-c}{g(x)}$ Where: $\lim_{x \to 0} f(x) = 0$, $\lim_{x \to 0} g(x) = 0$, $f,g \in C^{1}$ and $c \in \mathbb{R}$ ...
0
votes
3answers
1k views

Pointwise convergence of $f_n(x)=x^n$

Ran across this problem studying online: Show that $f_n(x)=x^n$ converges pointwise on the interval $x\in[0,1]$ and identify the limit function. Well if we we have $x=1$ we get $f_n(1)=1^n=1,\quad ...
1
vote
0answers
214 views

Equivalence of definitions for upper semicontinuity

I am trying to show that a function is upper semicontinuous if and only if the preimage of any open ray $(-\infty, a)$ is open. However, the definition that was given for upper semicontinuity is that ...
4
votes
1answer
145 views

Proving $\sup(|f|) - \inf(|f|) \leq \sup(f) - \inf(f)$

I appreciate if you could give me some hints on how to prove that : $$\sup(|f|) - \inf(|f|) \le \sup(f) - \inf(f)$$
4
votes
2answers
116 views

How to calculate this improper integral $\int_0^{+\infty} e^{-(ax+\frac{b}{x})^2}\mathrm{d}x$?

How to calculate this improper integral $$ \int_{0}^{\infty}{\rm e}^{-\left(ax\ +\ b/x\right)^2}\,{\rm d}x\ {\large ?} $$
1
vote
2answers
106 views

Continuous convergence fails

Given that $f_n \rightarrow f$ uniformly and that $x_n \rightarrow x$, then I was wondering whether $d (f_n (x_n), f(x)) < \epsilon$, that is, does $f_n(x_n) $converge to$ f(x)$? Any ideas? Been ...
0
votes
1answer
56 views

How to do trial and error for my model

I have the following Model: $(A-BX)^TS^{-1}(A-BX)$ which I would like to tune only $S$ and $B$ to make it equal to $mI$ where $A\in \mathcal{M}_{n,m}$, $B\in \mathcal{M}_{n,n}$, $X\in ...
1
vote
0answers
116 views

Characterization of Dirac Measure

Let $x_0$ be a point in a set $X$ and $\delta_{x_0}$ the Dirac measure concentrated at $x_0$. Characterize the nonnegative real-valued functions on $X$ that are integrable over $X$ with respect to ...
2
votes
1answer
369 views

Functions that are integrable with respect to counting measure

Let $\eta$ be the counting measure on the natural numbers, $N$. Characterize the nonnegative real-valued functions (that is, sequences) that are integrable over $N$ with respect to $\eta$ and the ...
-1
votes
1answer
79 views

On the space $C[a,b]$ of continuous functions on $[a,b]$

On the space $C[a,b]$ of continuous functions on $[a,b]$ prove that $du(f,g)=\sup|f(x)-g(x)|$ is a metric. Show that $f_{n}$ converges uniformly to $f$ on $[a,b]$ if and only if $\lim \,du(f_{n}, ...
0
votes
2answers
159 views

Real Analysis - bounded above (suprema)

The problem is attached as a picture.
2
votes
1answer
73 views

I need to show that the intersection of $T$ and $ \bar{I}$ is equal to empty as well.

Lets assume that $T⊂R^{n}$ is open and $I⊂R^n$ & $T\cap I=\varnothing$. I need to show that the intersection of $T$ and $\bar{I}$ (the closure of $I$) is empty as well. How to show this? I have ...
3
votes
1answer
95 views

The sum of Darboux is a Darboux function?

I was thinking last days about the following problem - the sum of Darboux is a Darboux function? Do You know a proof or counter example?
0
votes
1answer
84 views

$f$ is measurable iff its restrictions are measurable

Let $E$ be a measurable subset of $X$ and $f$ an extended real-valued function on $X$. Show that $f$ is measurable if and only if its restrictions to $E$ and $X\setminus E$ are measurable.
0
votes
1answer
109 views

Showing a function is continuously differentiable

Let $f=\begin{cases} 0 , \mbox{if x=(0,0)}\\ \frac{x^3}{x^2+y^2}\ \mbox{otherwise}\end{cases}$ Let $\gamma$ be a differentiable mapping from $\Bbb{R} \to \Bbb{R}^2$ with $\gamma(0)=(0,0)$ and ...
4
votes
1answer
124 views

The sum of infinite series $\sum_{k=1}^{\infty}2\sin\left(\frac{1}{k}-\frac{1}{k+1}\right)\cos\left(\frac{1}{k}+\frac{1}{k+1}\right)$

Determine the sum of the infinite series $$\sum_{k=1}^{\infty}2\sin\left(\frac{1}{k}-\frac{1}{k+1}\right)\cos\left(\frac{1}{k}+\frac{1}{k+1}\right).$$
1
vote
1answer
111 views

Convergence to function that is not measurable

Suppose ($X,\cal M, \mu$) is not complete. Show that there is a sequence {$f_n$} of measurable functions on $X$ that converges pointwise a.e. on $X$ to a function $f$ that is not measurable.
1
vote
1answer
131 views

$f = g$ a.e., $f$ is measurable, $g$ is not

Suppose ($X,\cal M, \mu$) is not complete. Let $E$ be a subset of a set of measure zero that does not belong to $\cal M$. Let $f=0$ on $X$ and $g=\chi _E$. Show that $f=g$ a.e. on $X$ while $f$ ...
4
votes
2answers
81 views

If $x\lt y $ for arbitrary real x and y there exists a real r $r$ such that $x \lt r \lt y$ and hence infinitely many.

If $x\lt y $ for arbitrary real $x$ and $y$ there exists a real r $r$ such that $x \lt r \lt y$ Prove that there is at least one r satisfying this inequality, and hence infinitly many. I was ...
0
votes
1answer
109 views

Use of Global picards theorem

Consider $y'=1+xy, y(0)=0 \:on\: [-b,b]$. Use the global Picards theorem to show there is a unique solution for any $b\lt \infty$. Ok so for $f,g$ that satisfy the conditions $\parallel Tf(x) - ...
4
votes
2answers
2k views

A function that is Lebesgue integrable but not measurable.

There is a theorem in our textbook that says, "Let $f$ be a bounded function on a set of finite measure $E$. Then $f$ is Lebesgue integrable over $E$ if and only if it is measurable." So I was ...
2
votes
3answers
199 views

Showing the boundary of the unit sphere is closed

I'm trying to show that this set is closed: $B=\{x\in \Bbb{R}^n \mid |x|=1\}$ Here is my proof : Let $y$ be a limit point then $B(y,r) \cap B \neq \emptyset $ then let $z\in B(y,r) \cap B \neq ...
2
votes
3answers
87 views

Which of the following is/are true:

Which of the following is/are true: $\log\dfrac{x+y}{2}\le\dfrac{\log x+\log y}{2}~\forall~x,y>0;$ ${e}^{\frac{x+y}{2}}\leq\dfrac{e^x+e^y}{2}~\forall~x,y>0.$
2
votes
1answer
209 views

Lebesgue and Borel Measurable

If a real-valued function on $R$ is measurable with respect to the $\sigma$-algebra of Lebesgue measurable sets, is it necessarily measurable with respect to the Borel measurable space ($R$, ...
1
vote
2answers
282 views

Continuously differentiable functions of bounded variation

From this question, we know that a continuous function of bounded variation is not necessarily absolutely continuous. But the example (Devil's staircase) given is not differentiable. What if we ...
2
votes
2answers
197 views

Power Series - domain of convergence

Determine the domain of convergence for each of the following power series: a) $\quad \displaystyle \sum_{k=1}^\infty (-1)^k \cdot \frac{x^{2k-1}}{(2k-1)!}$ b) $\quad \displaystyle ...
0
votes
1answer
338 views

approximating a riemann integrable function by sequences of step functions and sequences of continuously differentiable functions

Suppose that $f$ is Riemann integrable on $[0,M]$. How can I show that a) $f$ can be approximated uniformly by a sequence of finite step functions? and b) by a sequence of continuously differentiable ...
-6
votes
5answers
513 views

What are $(-1)^\infty$ and $1^{-\infty}$?

$$(-1)^\infty = ?$$ $$1^{-\infty} = ?$$ Is it possible to find such terms? I am very much confused by these problems.
5
votes
3answers
292 views

Prove that $\lim_{x\to 2}x^2=4$ using $\epsilon-\delta$ definition.

Prove that $\lim_{x\to 2}x^2=4$ using $\epsilon-\delta$ definition. By the mean of $\epsilon-\delta$ definition, $|x-2|\le \delta,|x+2|\le \delta+4$ then $|x-2||x+2|\le ...
2
votes
1answer
131 views

Justifying taking limit into an integral

I want to write the following equality: $$\lim_{h\to0}\left[\int\limits _{a}^{b}\frac{1}{h}\left[g\left(x-y+h\right)-g\left(x-y\right)\right]f\left(y\right)dy\right]=\left[\int\limits ...
2
votes
3answers
122 views

Example of a (dis)continuous function

The following thought came to my mind: Given we have a function $f$, and for arbitrary $\varepsilon>0$, $f(a+\varepsilon)= 100\,000$ while $f(a) = 1$. Why is or isn't this function continuous? I ...
1
vote
1answer
619 views

Some questions about functions of bounded variation: Jordan's theorem

I was trying to do some of these questions to check my understanding about the topic, but I'm not sure if they're correct. Here are my answers. 1) Suppose $f$ is continuous on $[0,1]$. Must there be ...
8
votes
4answers
428 views

limit of an integral of a sequence of functions

Suppose that $f$ is continuous on $[0,1]$. ($f'(x)$ may or may not exist). How can I show that $$\lim_{n\rightarrow\infty} \int\limits_0^1 \frac{nf(x)}{1+n^2x^2} dx = \frac{\pi}{2}f(0)\;?$$ My ...
2
votes
2answers
237 views

How to show product of two nonmeasurable sets is nonmeasurable?

How can I show that the cartesian product of a nonmeasurable set in $\mathbb R$ and a nonmeasurable set or a measurable set with nonzero measure in $\mathbb R$ is nonmeasurable? I have only learned ...
1
vote
1answer
52 views

Uniform convergence of a specific sequence of functions

Problem Studying for my finals currently and one of the my book's questions goes as follows, prove what you can about the uniform convergence of the following series by Theorem II (Weierstrass ...
1
vote
0answers
73 views

(Real Analysis) Integration of two functions

Note ($\Omega,A,\mu)$ is a finite additive space. Let $f\in \bar S(A, \mathbb{R})$ and $y\in Y$ where $Y$ is Banach space. Prove $\int _\Omega yf d\mu = y \int _\Omega f d\mu$. Also, for any $X\in A$, ...
2
votes
1answer
113 views

It is possible to construct such function?

It is possible to construct a $C^1$ function $f:\mathbb{R}\rightarrow\mathbb{R}$, such that $f(0)=0$; in every interval containing the origin, $f$ has infinite zeroes and ...
5
votes
0answers
74 views

Convergence of $\sum^n_{k=1}\frac1k$ after removing terms containing the digit $p$ [duplicate]

We know that $\sum^n_{k=1}\frac1k$ diverges. But if I were to pick a digit $p$ so that $p$ is an integer between $0$ and $9$ inclusive, and then I removed all terms in the sum $\sum^n_{k=1}\frac1k$ ...
1
vote
1answer
121 views

Cantor Function Question

I am looking for an explicit expression for the Cantor function for points in the cantor set. Does anyone know of one? Thanks
3
votes
2answers
177 views

Rectifiability of Hölder Continuous functions

If a curve is Lipschitz continuous, it is rectifiable. What can be said about Hölder Continuous ones? Are they rectifiable too? Is there either proof or else an example of a function that is Holder ...
0
votes
1answer
59 views

Two quick questions about convergence (in the context of pointwise vs. uniform convergence)

I found this example online: Let $\{f_{n}\}$ be the sequence of functions on $(0,\infty)$ defined by $f_{n}(x)=\frac{nx}{1+n^{2}x^{2}}$ .This function converges pointwise to zero. ...
1
vote
1answer
507 views

A proof of the the second derivative test?

Suppose $f\in C^3$ in some ball centered at a, where $a\in \Bbb{R}^2$,and $\nabla f=0$ at a, but not all second derivatives of $f$ are zero at a. Show how can local maximums local minimums or neither ...
6
votes
3answers
150 views

How to compute Integral

I wonder how to compute the following integral (for any natural number $n\geq 0$): $$\int_a^b (b-x)^{\frac{n-1}{2}}(x-a)^{-1/2}dx.$$ Does anyone know the final answer and how to get there? Is there a ...
4
votes
2answers
178 views

Rational approximations to $\sqrt 2$

I find this problem is very interesting, but now I can't solve it. Given $n$ a positive integer, let $$f(n)=\min_{m\in\Bbb Z}{\left\lvert\sqrt{2}-\dfrac{m}{n}\right\rvert}.$$ If there is a sequence ...
0
votes
2answers
118 views

Suppose that $x$ is a fixed nonnegative real number such that for all positive real numbers $\epsilon$, $0≤x≤\epsilon$. Show that $x=0$. [duplicate]

Suppose that $x$ is a fixed nonnegative real number such that for all positive real numbers $\epsilon$, $0≤x≤\epsilon$. Show that $x=0$.
1
vote
1answer
76 views

Lebesgue measurability of $f\circ g$

Let $f:\mathbb R\to \mathbb R$ be Lebesgue measurable and $g:\mathbb R\to \mathbb R$ be continuous. Do we also need extra conditions to ensure that $f\circ g$ is Lebesgue measurable?
1
vote
1answer
151 views

Borel and Lebesgue measurability of a continuous function

Let $f:[0,1]\to \mathbb R$ be continuous and suppose $f'$ exists on $(0,1)$. Is it true to say that $f'$ is Borel measurable? Or Lebesgue measurable?
4
votes
2answers
134 views

How to test the convergence of $\sum_{n=1}^\infty \dfrac{1}{\sqrt n}\tan\left(\dfrac{1}{n}\right)?$

How to test the convergence of$$\sum_{n=1}^\infty \dfrac{1}{\sqrt n}\tan\left(\dfrac{1}{n}\right)?$$ I'm clueless.
0
votes
0answers
27 views

Calculating assumed loss of traffic

If I have data reported monthly and suddenly there was a change $X$ to the monthly reported data. How do I calculate to solve for the actual difference or loss that may exist. To be more specific this ...
2
votes
2answers
111 views

a problem of existence of local maxima and minima

Let $k:[0,1]→\mathbb{R}$ be a continuous function. Suppose that $k$ has local maximum at two distinct points $x_1 , x_2$ in $[0,1]$. show that $k$ has a local minimum at some point $x_3$ in [$0,1$]. ...