Okay, what have you tried? This really only involves verifying the definitions of "homomorphism" and "subgroup".
The only condition on a "homomorphism" is that it maintain group operations. If f and g are two differentiable functions, is it true that theta(f+ g)= theta(f)+ theta(g)?
Since saying that C1 is a group means that things like associativity of the operation is true, we don't have to reprove those for operations in the subgroup. We only need to show
(1) the 0 element is in the subset
(2) the subgroup is "closed" under the group operation
The 0 element of this group is the 0 function, f(x)= 0 for all x. Is that in this subset?
Suppose f and g are in this subset. Is f+ g also in this subset? That is, is (f+ g)(0)= 0?