continuous onto function from (0,1)x(0,1) to [0,1]x[0,1]

Question

How do I define such a function between the open unit square onto the closed unit square? I cannot think of any definition for a function that will be surjective.

Answer 1

Map (0,1)x(0,1) to (0,2)x(0,2) by multiplying by 2 and then truncate everything outside (0,1)x(0,1) to its support.

Proving it is a matter of cycling through the lemmas and theorems you are allowed to use. For example,

• linear functions are continuous

• continuous in R2 metric space is topologically continuous

• constant functions are continuous

• restrictions of continuous function are continuous

• composition of continuous functions are continuous

• truncating is constant function and identity function respectively over two disjoint subsets; in other words, truncating is the composition of two restrictions of two respective continuous functions

You may find it easier to write out the truncation part using first principle.