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Unformatted text preview: MATH 8, SECTION 1, WEEK 5 - RECITATION NOTES TA: PADRAIC BARTLETT Abstract. These are the notes from Friday, Oct. 29th’s lecture. In this talk, we discuss the Weierstrass function, an example of a function that is continuous everywhere and differentiable nowhere. First: a warning: today’s lecture was utterly ridiculous! Don’t be afraid if it’s confusing in parts, or complicated; the math going on here is really tricky! This result is typically presented in a graduate-level course in analysis; I wanted to show you guys that what you’re doing in Math 1a is in fact real math, and that you can prove and study some seriously complicated things with the tools you have. So, if you’re reviewing for a midterm, don’t worry about this lecture too much! Think of this as a cross between an illustration of what you *can* do with your current tools, and a demonstration of a ton of proof techniques we’ve used/will use again throughout the course (with some aspects of an advertisement for advanced maths sprinked in!) 1. Random Question Question 1.1. Suppose you have a Z × Z grid of squares. Consider the following game we can play on this board: • Starting position: place one coin on every single square below the x-axis. • Moves: If there are two coins in a row (horizontally or vertically) with an empty space ahead of them, you can “jump” one of the coins over the other – i.e. you can remove those two coins and put a new coin on the space directly ahead of them. In this game, how “high” on the y-axis can you get a coin? Can you get one to height 3? Higher? Why or why not? 1 2 TA: PADRAIC BARTLETT 2. The Weierstrass Function Today’s lecture is centered around answering the following question: Question 2.1. Is there a function that is • continuous on all of R , yet • differentiable nowhere on R ? As it turns out, the answer is yes! Specifically, consider the following function: f ( x ) = ∞ X n =1 cos(101 n · πx ) 2 n . Functions of this form (where 2 and 101 can be replaced with different constants a,b , that satisfy some set of properties) are called Weierstrass functions. A graph of a Weierstrass function, taken from Wikipedia, is shown below: We will prove that our function has the claimed properties in three parts: Claim 2.2. The Weierstrass function f ( x ) is defined – i.e. the series ∑ ∞ n =1 cos(101 n · πx ) 2 n converges – for any value x ∈...
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