# Linked Questions

2answers
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### 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 ...
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'...
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 ...
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 ...
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 ...
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!
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 ...
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 ...
1answer
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1answer
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### 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 ...
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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