Logically inconsistent optimality condition
There is $C_t$ the number and $C_t()$ the function. This is already so in your first equation. Your second equation says that the function is constant over $[0,1]$. In which case it is true that $C_t$ the number equals $C_t()$ the function's value times 1. 

The same thing with a different notation:
$$ C_t \triangleq \int_0^1 f_{C,t}(i) di.$$
If there exists a constant $c_t$, where for all $i$
$$f_{C,t}(i) = c_t.$$
then
$$ C_t \triangleq \int_0^1 f_{C,t}(i) di = \int_0^1 c_t di = c_t. $$