Sets and real numbers

Here is the problem sheet.

Topology

Here is the summary of basic topological notions. Here are the problems.

Sequences and series

Here are the problems. Here are some examples of partial sums. Here are the notes covering the limit superior and inferior.

Here is a visualization of the proof of the Heine-Borel theorem. The explanation and the proof are in the description of the video.

Continuous functions

The problem sheet. The function sin¹⁄_x with an essential discontinuity at 0.

Differentiation and integration

Here is the sequence of functions xⁿ|x| which are increasingly more smooth. Here is the sequence of oscillating functions xⁿsin¹⁄_x which are also increasingly more smooth.

If the sequence defined recursively by x_n=f(x_n-1) converges, then its limit is a fixed point of continuous function f. We can visualize the convergence process by plotting the points (x₀,x₁), (x₁,x₂), (x₂,x₃), (x₃,x₄),... in the plane:

• fixed point of the sine function
• fixed point of the cosine function

Here you can explore Riemann sums. Here is an example of an improper integral which converges but not absolutely.

Uniform convergence

Examples of series of functions ∑sinⁿx and ∑sin xⁿ.

Here are the first few terms of a certain sequence of curves f_n:[0,1]→ℝ². This sequence converges uniformly to a continuous function f:[0,1]→ℝ². The image of f is the full square [0,1]². Here is a more detailed explanation. Here are some additional examples of uniformly convergent sequences of curves. Here is a higher dimensional example.