I taught Analysis I (Math 3150) in Fall 2016 at UConn. Interestingly, I never took analysis as an undergraduate (I jumped straight into measure theory instead since it sounded more interesting). So I learned quite a lot teaching this course! We used the textbook “Understanding Analysis” by Stephen Abbot. I highly recommend this book. I would also highly recommend complementing this with “How to Think About Analysis” by Lara Alcock. In addition, I wrote supplementary notes to complement the course textbook. Click here for these notes. Warning: They should go through some editing process.
A sequence of functions that converges to an everywhere continuous but nowhere differentiable function (done in Mathematica)
The corresponding sequence of derivatives showing more dramatically why the limit of the sequence from above is nowhere differentiable (done in Mathematica)