The assignment went reasonably well, but there are a few things worth commenting on (if you are impatient, the link to the solution is on the bottom of the page):
Problem 1: This was meant as a routine problem, but many found it challenging. The most common mistake was giving a proof based on d(x,y) = |x-y|, but this relationship only holds for the standard metric on R (or a metric derived from a norm if |·| denotes a norm).
Problem 2a): This problem went well, but some used an unnecessarily complicated approach. When asked to show that fn converges pointwise to f, if suffices to show that lim n->∞ fn(x)=f(x) for all x, and you are allowed to use all the methods you have learnt in calculus, such as Hôpital's rule (but you can only use these methods when the functions are taking values in R).
Problem 2b): Some solved this problem by proving that f'(x) approaches different limits when x approaches 1 from above and below, but this doesn't in itself prove that f'(1) does not exist, only that it cannot be continuous at 1. It is better to use the definition of the derivative at x=1.
Problem 2c): This was a rather disappointing point which showed that many do not have a good grasp of the distinction between pointwise and uniform convergence. Uniform convergence is a central concept in the course, and problems involving uniform convergence have often been given as "easy" exam problems.
Problem 3a): I thought the idea of reparametrization would be an easy one after MAT1110, but we got surprisingly many questions about this point.
Problem 3b): This point when relatively well, but many struggled with the interplay between intuition and formal mathematics. Intuitively, a path is just a way of getting from a point to another, but formally it is a function from an interval [a,b] into X with certain properties. When you are solving a problem, you should let your intuition lead you, but the solution has to be formulated in terms of the mathematical definitions.
Problem 3c): Many missed the main point which is that one has to prove that the path lies in P.
Problem 3d): Some made this problem unnecessarily complicated. It is usually a good idea to keep to a simple strategy. If you want to prove that a set C in included in another set P, pick an arbitrary point in C and explain why it has to belong to P.
Problem 3e): Many had the right idea, but not all found a good way of expressing it. Again it is a good idea to start from basics: If you are to prove that a set f(C) is path-connected, you start with two (arbitrary) points in f(C) and describe how you can find a path connecting them.
Problem 3f): This was the challenge of the assignment, and those who got it right, may be proud of themselves. Note that there is no reason why r(s) should be y even if s=inf{t : r(t)=y}. It may well be x, but both possibilities lead to a contradiction.
Problem 3g:) Many had the right idea. It is convenient to construct the example such that f-1(D) becomes a two-point set as you can then just appeal to part f) to prove that it is not path-connected.