Linked Questions

2
votes
2answers
5k views

Why do non-Decreasing Functions have countable discontinuities [duplicate]

I was reading some notes and one of the results in it implicitly used a result which fell along the lines of "non-decreasing functions have countable discontinuities". I don't completely understand ...
4
votes
2answers
1k views

Why are discontinuities of monotonic $f : (a, b) \to \mathbb R$ countable? [duplicate]

Possible Duplicate: Showing properties of discontinuous points of a strictly increasing function How to show that a set of discontinuous points of an increasing function is at most countable I'...
0
votes
1answer
76 views

How to see that this set is countable? [duplicate]

Let $f : \mathbb{R}\to \mathbb{R}$ be a monotone function and consider the set $S$ of all points $a\in \mathbb{R}$ such that $$\lim_{x\to a^-}f(x)\neq \lim_{x\to a^+}f(x).$$ I want to show that ...
11
votes
2answers
5k views

Is there an everywhere discontinuous increasing function?

Does there exist a function $f : \mathbb{R} \rightarrow \mathbb{R}$ that is strictly increasing and discontinuous everywhere? My line of thought (possibly incorrect): I know there are increasing ...
10
votes
4answers
3k views

Is the set of discontinuity of $f$ countable?

Suppose $f:[0,1]\rightarrow\mathbb{R}$ is a bounded function satisfying: for each $c\in [0,1]$ there exist the limits $\lim_{x\rightarrow c^+}f(x)$ and $\lim_{x\rightarrow c^-}f(x)$. Is true that the ...
7
votes
2answers
561 views

Two monotone functions which equal on rational numbers

Let $f,g:\mathbb R\to \mathbb R$ be increasing and $f(r)=g(r)$ for every $r\in\mathbb Q$. Must we have $f(x)=g(x)$ for every $x\in\mathbb R$? Thanks in advance!
8
votes
4answers
336 views

if $f:[0,1] \to \mathbb{R}$ is increasing, show that $f$ is the pointwise limit of a sequence of continuous functions over $[0,1]$ [duplicate]

if $f:[0,1] \to \mathbb{R}$ is increasing, show that $f$ is the pointwise limit of a sequence of continuous functions over $[0,1]$ Intuitively this makes sense but I am having trouble with showing ...
3
votes
4answers
1k views

Theorems about functions with uncountable number of discontinuities

I have seen a nice number of theorems that start with "suppose that $f$ is continuous function" or with some equivalent claim and then, with only that, or with some additional assumptions some theorem ...
4
votes
1answer
2k views

Showing properties of discontinuous points of a strictly increasing function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be strictly monotonically increasing. (i) Is $f$ not continuous at $p \in \mathbb{R}$, there exists a non-empty, open interval $(a_p, b_p) \subset \mathbb{R}...
6
votes
1answer
1k views

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable?

If $g$ is Riemann-integrable in a closed interval and $f$ is a increasing function in a closed interval, is $g\circ f$ Riemann-integrable? To clarify: the problem stated that the composition is well ...
8
votes
1answer
269 views

Prove that the set of all monotone functions on $[0,1]$ has same cardinality as $\mathbb R$

I am having difficulty answering the following question on "Notes on Set Theory" by Moschovakis, 1st edition: Prove that the set $K$ of all monotone real functions on the closed interval $[0,1]$ ...
7
votes
2answers
209 views

Monotone function with $f(\mathbb{R}) = \mathbb{R} \backslash \mathbb{Q}$

I want to prove per contradiction, that there doesn't exist a strictly monotone function $f:\mathbb{R} \to \mathbb{R}$ with $$ f(\mathbb{R}) = \mathbb{R} \backslash \mathbb{Q} $$ but I'm not sure if ...
3
votes
2answers
155 views

Why can we 'choose' continuity points?

Let $F$ and $F_n$ be distribution functions with $\lim_n F_n(x)=F(x)$ for all continuity points $x$ of $F$. In a proof there is the following part: Block quote [...] choose the finite points $a=...
0
votes
1answer
340 views

A question on the “sum” of an uncountable “number” of positive quantities [duplicate]

In the answer to this post Ittay Weiss wrote that, "...a sum of infinitely many positive elements can be bounded only if there are countably many elements." Though I asked about a rigorous proof of ...
5
votes
1answer
126 views

Which types of convergence preserve this property?

Say, we have a sequence of random variables with $X_n \geq 0$ almost everywhere. Which of the following types of convergence: almost everywhere: $X_n \xrightarrow{a.e.} X$ in probability: $X_n \...

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