The Analysis in Hilbert space notes (MA40256) from 2017/18 are available here: MA40256.

Analysis 1B (MA10207)

Feel free to email me with any questions or queries, or for any typos:

Some more epsilon-delta examples (from tutorials and extra): examples.
Here are the solutions: example_solutions.

Here are some definitions of limits at infinity: limits_at_infinity.
There are also orders of growth on this sheet (big O and little o notation).

Here is the example of continuity of $f(x) = x^{-3}$ on $(0,2)$ from week 3: continuity_example.
Here is another example of continuity on an interval from week 4: continuity_example_2.

Here is a short introduction to open/closed sets and density in $\mathbb{R}$: open_closed
For a more indepth discussion, see MA20218 (Analysis 2A), in particular metric spaces.

The following proves Taylor's theorem with Lagrange and Cauchy error term: taylors_theorem_IVT.
This document also includes the statement of the inverse function theorem, as in PS6 H4.

Below are solutions to the "tutorial" questions on the problem sheets.
Solutions 0: solutions0.
Solutions 1: solutions1.
Solutions 2: solutions2.
Solutions 3: solutions3.
Solutions 4: solutions4.
Solutions 5: solutions5.
Solutions 6: solutions6.
Solutions 7: solutions7.