All Questions
19
questions with no upvoted or accepted answers
4
votes
0
answers
127
views
Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 9,592
3
votes
0
answers
56
views
Discontinuous function of two variables
Let \begin{equation}
f: \mathbb{R}^{2} \rightarrow \mathbb{R}
\end{equation} be a function of two real variables given by
\begin{equation}
f(x,y) = \begin{cases}
\frac{x}{y} & \text{...
- 61
2
votes
0
answers
62
views
What is the precise definition of infinite limit at infinity of a function?
I saw the following definition on youtube.
Let $f$ be a function defined on some interval $(a, +\infty)$
$\lim\limits_{x \to +\infty} f(x)= +\infty$ (1)
means $\forall N > 0$ $\exists M > 0$ ...
- 33
1
vote
0
answers
26
views
Finding functions using cosine
Assume that the sine and cosine functions are continuous at the point 0.
(a) Find functions $a(x), b(h), c(x), d(h)$ such that $b(h)$ and $d(h)$ are continuous at $h = 0$ and $b(0) = d(0) = 0$, such ...
- 139
1
vote
1
answer
151
views
Showing that Derivative is Linear
Question is: In $$f(a+h) - f(a) = h f'(a + \frac h 2), \qquad a, h \in \mathbb R$$ show $f'$ is line.
I have no problems with the first part. I'm however having trouble with taking the derivative ...
- 1,522
1
vote
0
answers
279
views
Generalize Squeeze Theorem
Theorem. Let $I$ be an interval having the point $a$ as a limit point. Let $g$, $f_1$, $f_2$ ,..., $f_n$ and $h$ be functions defined on $I$, except possibly at $a$ itself. Given that for every $x$ in ...
- 5,872
1
vote
1
answer
50
views
Open question about limits of functions
I came across something today that to my surprise I had never seen before. Basically I had a sequence of continuous functions $f_i \rightarrow f$, all from $[0,1]$ to $\mathbb{R}$ and a convergent ...
- 21
1
vote
0
answers
37
views
Finding a closed mathematical form for a parametrized function series
I am dealing with the following parametrized function series defined by
$$
F(x) := \sum_{n=0}^{\infty} 2\left( \varphi_n(x)-(n+1)(n+2)x^2 \psi_n(x) \right)x^{n+1} \, ,
$$
where $x \in [0,1)$.
The ...
1
vote
2
answers
69
views
Uniqueness of limiting functions
Let $\{f_n\},\ f_n:[a,b]\mapsto\mathbb{R}$. Suppose that the limiting function, $f(x)$, of the sequence satisfies
$$
\lim\limits_{n\rightarrow\infty}\int_a^b|f_n(x)-f(x)|^2dx = 0.
$$
Is this limiting ...
- 73
1
vote
1
answer
111
views
Integration of characteristic function with varying boundaries
I'm a bit puzzled about integrals with indicator/characteristic functions in them. How do I start computing the following integrals?
$$
A\int_{-\infty}^{\infty}f(x)\chi_{[-a+x,a+x]}dx
$$
and
$$
A\int_{...
- 11
1
vote
1
answer
108
views
Limit vs interior definition of continuity
Suppose I have two topological spaces $X$ and $Y$ whose topologies are defined by interior operators $\text{int}_X$ and $\text{int}_Y$ respectively, as well as a function $f$ with domain $I$ (for ...
0
votes
0
answers
44
views
About the definition of the limit of a function ("Mathematical Analysis 2nd Edition" by Tom M. Apostol)
4.5 LIMIT OF A FUNCTION
In this section we consider two metric spaces $(S, d_S)$ and $(T, d_T)$, where $d_S$ and $d_T$ denote the respective metrics. Let $A$ be a subset of $S$ and let $f:A\to T$ be a ...
- 8,750
0
votes
0
answers
61
views
Jump Continuity of the function $f(x) = \frac{ax+b}{cx+d}$
With trying with examples, I found that
the function $$f(x) = \frac{ax+b}{cx+d}$$ has a jump continuity when $a\cdot \frac{-d}{c} + b = 0$ at point
$(-d/c, \frac{a+b}{c+d})$
However, I could not find ...
- 1
0
votes
0
answers
76
views
Is my answer to this limit correct?
Define a function $ z(x, y) = \frac{11x^3+5y^3}{x^2+y^2} $. Using the continuous extension, what is the value of the following limit:
$$ \lim_{(\Delta x, \Delta y) \rightarrow (0, 0)\\ \text{along the ...
0
votes
0
answers
31
views
Continuous extension of $f(x,y)=\left(\frac{1+y+x^2}{y}\right)^y$
The function $f(x,y)=\left(\frac{1+y+x^2}{y}\right)^y$ is not defined for $y=0$, so to extend $f$ continuously I want to define $f(x,0)$ as :
$$f(x,0)=\lim _{y \rightarrow 0} \left(\frac{1+y+x^2}{y}\...
- 918