I will be discussing continuity and the popcorn function. First the basic definition from pre-calc, a function you can draw smoothly without lifting up your pen. You can all picture these kinds of functions. There exists a function that breaks this intuition. It is continuous for all of the irrational numbers, but not the rational numbers, and it does not look like a smooth line with no breaks. This is the Popcorn Function, which is what I will be discussing in my talk.
I looked up a graph of the popcorn function. It looks very confusing to have a function that could be graphed like that. I am intrigued to see the development of continuity with the definition that we now know from Real Analysis. It will be very interesting to connect different definitions to view this function. I am excited to hear the explanation of this topic!
Your talk on the popcorn function sounds very interesting! I have not heard of this function before, so I am interested in learning more about the function behaviors and properties. I look forward to hearing more about why it is continuous for irrational numbers but not rational numbers. Also, I think it is really cool that you chose a topic that builds on our class content from Real Analysis! I am eager to hear more about this application of our class focus on continuity!
This topic sounds very intriguing. Figuring out continuity in Real Analysis was difficult to understand, so I’m looking forward to seeing how you describe it with a much more complicated sounding function. I have not had much exposer to theories involving rational numbers that I can recall, so it’ll be interesting to see how they work here.
Although Real Analysis was not my strong suit (as we all know) I did find this function interesting and how it relates to Real Analysis. Similarly to as you mentioned, it helped open my eyes to why definitions such as the one using epsilon and delta we used in Real Analysis are essential to understanding math at a deeper level. The graph itself is also pretty cool!