The Math Kid
A collection of mathematical fascinations, not strictly limited to the subject itself but also people's interactions and interpretations. My curiosity for the subject is infinite, but I try to stay away from the sciences. The thrill lies in the purity, not the application.
Info on my icon can be found here.
Seriously who thinks of these questions. Like its not hard, but the way it’s written. A “microwave cooking enthusiast” like who the fuck thinks of these questions
![contemplatingmadness:
Thomae’s function, named after Carl Johannes Thomae, also known as the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function,[1] the Riemann function or the Stars over Babylon (by John Horton Conway) is a modification of the Dirichlet function. This real-valued function f(x) is defined as follows:
If x = 0 we take q = 1. It is assumed here that gcd(p, q) = 1 and q > 0 so that the function is well-defined and non-negative.
Discontinuities
The popcorn function is perhaps the simplest example of a function with a complicated set of discontinuities: f is continuous at all irrational numbers and discontinuous at all rational numbers.
Informal Proof
Clearly, f is discontinuous at all rational numbers: since the irrationals are dense in the reals, for any rational x, no matter what ε we select, there is an irrational a even nearer to our x where f(a) = 0 (while f(x) is positive). In other words, f can never get “close” and “stay close” to any positive number because its domain is dense with zeroes.
To show continuity at the irrationals, assume without loss of generality that our ε is rational (for any irrational ε, we can choose a smaller rational ε and the proof is transitive). Since ε is rational, it can be expressed in lowest terms as a/b. We want to show that f(x) is continuous when x is irrational.
Note that f takes a maximum value of 1 at each whole integer, so we may limit our examination to the space between and . Since ε has a finite denominator of b, the only values for which f may return a value greater than ε are those with a reduced denominator no larger than b. There exist only a finite number of values between two integers with denominator no larger than b, so these can be exhaustively listed. Setting δ to be smaller than the nearest distance from x to one of these values guarantees every value within δ of x has f(x) < ε.](http://25.media.tumblr.com/tumblr_m32lonrc6E1qcyo2po1_400.png)
Thomae’s function, named after Carl Johannes Thomae, also known as the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function,[1] the Riemann function or the Stars over Babylon (by John Horton Conway) is a modification of the Dirichlet function. This real-valued function f(x) is defined as follows:
If x = 0 we take q = 1. It is assumed here that gcd(p, q) = 1 and q > 0 so that the function is well-defined and non-negative.
Discontinuities
The popcorn function is perhaps the simplest example of a function with a complicated set of discontinuities: f is continuous at all irrational numbers and discontinuous at all rational numbers.
Informal Proof
Clearly, f is discontinuous at all rational numbers: since the irrationals are dense in the reals, for any rational x, no matter what ε we select, there is an irrational a even nearer to our x where f(a) = 0 (while f(x) is positive). In other words, f can never get “close” and “stay close” to any positive number because its domain is dense with zeroes.
To show continuity at the irrationals, assume without loss of generality that our ε is rational (for any irrational ε, we can choose a smaller rational ε and the proof is transitive). Since ε is rational, it can be expressed in lowest terms as a/b. We want to show that f(x) is continuous when x is irrational.
Note that f takes a maximum value of 1 at each whole integer, so we may limit our examination to the space between
and
. Since ε has a finite denominator of b, the only values for which f may return a value greater than ε are those with a reduced denominator no larger than b. There exist only a finite number of values between two integers with denominator no larger than b, so these can be exhaustively listed. Setting δ to be smaller than the nearest distance from x to one of these values guarantees every value within δ of x has f(x) < ε.
Found this on Tumblr a while ago. Didn’t reblog and subsequently lost it. Glad to have finally found it again!
A shape that can be dissected into smaller copies of the same shape is called a reptile or rep-tile.
Using an orange to illustrate an Euler spiral.
About
Recreational mathematician and aspiring teacher. Grew up a math nerd and consider the subject a major part of my identity. Fascinated by just about everything. Here's hoping this blog will open me up to some more creative outlets. That might be good for me at this juncture in my life :)
