Lecture 18 Weierstrass's Example of a Continuous and Nowhere Differentiable Function
[SQUEAKING] [RUSTLING] [CLICKING] CASEY RODRIQUEZ: So we're going to continue with our discussion of the derivative. So now, let me recall the definition we introduced at the end of last time of the derivative. So let I be an interval, meaning it could be open, closed, it could go out to plus infinity, it could go out to minus infinity. But you know what an interval is. And let's take a function from that interval to R. So we say, f is differentiable at a point c in I. If this limit of f of x minus f of c over x minus c, this difference quotient, if this limit exists. If this limit exists, we also denote it by f prime of c. And last time, we showed that-- we gave a simple example, that from last time, that if x equals alpha times x to the n, then this function is differentiable at every c, and its derivative is equal to n times alpha x c to the n minus 1. Now, let's state this in an equivalent way using sequences. Remember, we had this characterization of limi...