Question

Consider the set of all functions [0,1] which are discontinuous. check if this set is a vector space over R with respect to pointwise addition and scalar multiplication

Answer 1

It is necessary to verify axioms of a vector space.
The verification is direct.
For instance,
If f,g : [0,1]->R are two functions and T is a real number, then
& f+g& and T*f
are also a functions [0,1]->R.