help pls (vector space and linear algebra)

[bright spruce]

i have to show both implications

[warped shuttle]

I apologise for being here, not to answer questions, but to ask further question, but I'm confused myself,
I don't understand why the (=>) is true in the first place,

if I take the constant function f such that f(x) = 3 for instance
(1) f is in E because f(x+1) = f(x) = 3
(2) f is in Im(Delta) because f"(x) = (f'(x))' = (0)' = 0

But one of the primitives of f is 3x and its integral over 0 and 1 is 3

or have I missed anything, any hypothesis ?

[bright spruce]

hey i managed to do both implications i will explain later if you want i have to go to class now tho

[warped shuttle]

Okay no sweat, whenever you can

[bright spruce]

f(x+1)-f(x) = 0

why is is equal 3