Is there any other way to make a $f :[0,1]\to \mathbb{R}$ discontinuous at every point of its domain?

Question

The Dirichlet function: $$f(x) = \begin{cases} 0, & \text{if $x$ is rational} \\ 1, & \text{if $x$ is irrational } \end{cases}$$ is an example of a function that is discontinuous everywhere on its domain , but a question that came to my mind is there any other function ?

Let $S$ be a dense set on $[0,1]$ such that $[0,1]- S$ is still a dense set,

$$g(x) = \begin{cases} 0, & \text{if $x \in S$} \\ 1, & \text{if $x \not\in S$ } \end{cases}$$ This function will do the same trick as Dirichlet function.

My question is : Is there any way a function $f:[0,1] \to \mathbb{R}$ could be discontinuous everywhere on $[0,1]$ without choosing a dense set $S$ such that such that $[0,1]- S$ is still dense and assigning two functions that don't intersect each other in $[0,1]$ to each one of them ? (for example : $$f(x) = \begin{cases} x, & \text{if $x \in S$} \\ \tan(x)+1, & \text{if $x \not\in S$ } \end{cases}$$ is the same as the Dirichlet function 2 disjoint dense sets and for each one of them we assign a function that these function don't intersect each other if there domain with the whole $[0,1]$)

I don't think that is possible for such function to exist because it had to "jump" back and forth infinity many times but I couldn't prove that.

There are MANY other nowhere continuous functions. For a fun example check out Conway's base 13 function. — stochastic-conch, Apr 03 '24 at 07:14
For example, choose $3$ disjoint sets, each of them dense, and choose functions for each? — davidlowryduda, Apr 03 '24 at 07:28
The functions you're defining have graphs that essentially look like the graphs of two non-intersecting functions, and thus unless at least one of the two functions is itself wildly discontinuous (and your two functions are not -- your 1st example is essentially two horizontal lines, and your 2nd example is a line along with essentially the tangent function's graph), the combined graph is not very spread-out in the plane. There are functions whose graphs are dense subsets of the plane. (continued) — Dave L. Renfro, Apr 03 '24 at 10:20
See Graph of discontinuous additive function is dense in $\mathbb R ^ 2$ AND Does there exist a function $f:[0,1] \to[0,1]$ such its graph is dense in $[0,1]\times[0,1]$? AND Function with range equal to whole reals on every open set AND Function whose image of every open interval is $(-\infty,\infty)$ (continued) — Dave L. Renfro, Apr 03 '24 at 10:20
AND Is there a function $f\colon\mathbb{R}\to\mathbb{R}$ such that every non-empty open interval is mapped onto $\mathbb{R}$? AND Can we construct a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that it has intermediate value property and discontinuous everywhere? AND graph is dense in $\mathbb{R}^2$ AND Is there a different name for strongly Darboux functions? (continued) — Dave L. Renfro, Apr 03 '24 at 10:21
AND this answer to Examples of dense sets in the complex plane. See also these sci.math posts: 5 October 2006 AND 6 October 2006. — Dave L. Renfro, Apr 03 '24 at 10:27

Is there any other way to make a $f :[0,1]\to \mathbb{R}$ discontinuous at every point of its domain?

0 Answers0