2025 on Tumblr: Trends That Defined the Year
I'd rather be in outer space 🛸
Kiana Khansmith
sheepfilms
cherry valley forever
oozey mess
izzy's playlists!
Aqua Utopia｜海の底で記憶を紡ぐ
macklin celebrini has autism
Mike Driver
Xuebing Du
#extradirty
Sweet Seals For You, Always
h
titsay
Peter Solarz
hello vonnie
Not today Justin
Misplaced Lens Cap
will byers stan first human second
seen from Syria
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@my-pink-code
The mouth speaks what the heart is full of. In this case, it appears to be a unit circle. <3 It’s an art experiment.
- Anjuli / Mathelle
Vector art credit: Webysther (Wikimedia Commons) Image is distributed under the terms of the Creative Commons Attribution ShareAlike License
Proof for the popcorn function’s integrability
Seems like I got a rush of new followers after my Dirichlet’s function and popcorn functions posts, which is very cool. It’s nice to see such a lively math community here on Tumblr. Thanks for connecting with me and welcome on board! *waves* I was requested proof for the popcorn function’s integrability on [0,1], so here it is.
The proof is university-level mathematics, so if you’re in high school, don’t be discouraged at all if it’s difficult to understand. And even if you’re at university, I think it’s normal to struggle with proofs like this if you’re just starting out. (Btw, for people struggling with or getting prepared for university-level analysis, I highly recommend Lara Alcock’s book How to Think About Analysis, which I will probably review later on.)
Let us be reminded that Thomae’s function was defined like this:
And here’s what it looks like on [0,1] when graphed:
Now here’s the proof for Thomae’s function’s Riemann integrability.
Let us remember Riemann’s condition:
We have chosen an arbirtrary epsilon that is greater than 0 and shown that for this epsilon, there exist such step functions g and h that g is less than or equal to f and h is greater than or equal to f on [a,b] and the integral of h over [0,1] minus the integral of g over [0,1] is less than epsilon.
Thus, according to Riemann’s condition, Thomae’s function is Riemann integrable on [0,1].
- Anjuli
I found a curious complex plane in my tea cup when studying for my analysis exam.
Vector art credit: Loadmaster (David R. Tribble) (Wikipedia) Photography is mine and the colors of the original vector art may have been changed. Image is distributed under the terms of the GNU Free Documentation License and the Creative Commons Attribution ShareAlike License
- Anjuli
Dirichlet’s function is nowhere continuous and nowhere differentiable. It is also nowhere Riemann integrable since its upper integral and lower integral do not equal anywhere.
The popcorn function, also known as Thomae’s function, is continuous at all irrational numbers and discontinuous at all rational numbers. The function is Riemann integrable on [0, 1] and its integral over [0, 1] equals 0. (If someone wants proof for this, I ca post it later on.)
Image credit: Smithers888 (Wikimedia Commons). The work is distributed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
mathelle
mathelle
Watercolor. It is said to be the most beautiful equation.
mathelle
mathelle
mathelle
mathelle
mathelle
www.mathelle.com
mathelle
The endlessly zooming golden spiral. I made this with Processing back in 2013 (or something). That was one of my proudest Tumblr moments, as this little spiral received thousands of likes very quickly. :)
Btw, Processing is a really nice tool for creative coding and making math visualizations.
mathelle
www.mathelle.com
