Continuity of stochastic integration with respect to path

Question

Let's consider the following question: Let $(W_t, t \in [0,1])$ be a standard Brownian motion and $f \in C[0,1]$ deterministic (or more generally $f \in L^2 [0,1]$). Define the stochastic process $(I_t, [0,1])$ by Ito integral $$I_t = \int_0^t f(s) d W_s.$$ Is there always a continuous map $F: C[0,1] \to C[0,1]$, such that $(I_t, \ t\in [0,1]) = F((W_t,\ t\in [0,1])$? Note that this is true when $f \in C^1$, as we have integration by part formula: $$I_t = f(t) W(t) - \int_0^t W(s) f'(s) ds.$$