Hi Scholars!

This week, a post on Stack Exchange caught my attention. It was an inquiry about pointwise convergence of functions, which we can define as follows.

A sequence of functions {fₙ} converges pointwise to a function f provided, for each point x, lim fₙ(x) = f(x).

Two notes:

It is likely counterintuitive for students to grasp that each function in the sequence shown in the above graphic obtains the maximum value of one, but the sequence still converges pointwise to the zero function. The key point is to emphasize the first step is to fix a point x in [0,1]. Then the limit is to be taken of the sequence {fₙ(x)} with respect to n.

Even though {fₙ} converges pointwise, the fact that each fₙ always has a maximum value of 1 causes {fₙ} to

*fail*to converge*uniformly*to the zero function. This difference between pointwise and uniform convergence is why I enjoy using tent functions (and related variants) to discuss convergence of functions.

That’s all for now. If you have any suggestions for other illustrations, especially on pointwise and/or uniform convergence, let me know!

Stay Awesome.

Howard