Digital Option with Volatility Smile

In the setting of the Breeden-Litzenberger relation, suppose now that the implied volatility depends on the strike, i.e. $\sigma = \sigma(K)$, so that the call price is given by:

for a smooth function $f$.

Question: Express $D_t(K)$ in terms of the partial derivatives of $f$, the implied volatility $\sigma(K)$, and its derivative $\sigma'(K)$.