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 ...
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'...
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 ...
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 ...
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 ...
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!
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 ...
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 ...
2k views

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 $(a_p, b_p) \subset \mathbb{R}... 1answer 1k views ### If$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$Riemann-integrable? To clarify: the problem stated that the composition is well ... 1answer 269 views ### Prove that the set of all monotone functions on$[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]$... 2answers 209 views ### Monotone function with$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 ... 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=...
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 \...