How to proof B&S pricing formula is continuous in time $t$ (or it is not?).
The general pricing formula is $$ C_t = e^{-r(T-t)} \mathbb{E}^*[(S_T-K)^+ | \mathcal{F}_t] \hspace{1cm} 0\leq t\leq T $$ Then for time at maturity $t=T$ $$ C_T = \mathbb{E}^*[(S_T-K)^+ | \mathcal{F}_T] = (S_T-K)^+ $$ which is logic. For other anterior time $t<T$, the integration calculation give $$ C_t = S_t \mathcal{N}(d_+) - e^{-r(T-t)} K \mathcal{N}(d_-) $$ with $$ d\pm = \frac{\text{ln}\frac{S_t}{K} + (r\pm\frac{\sigma^2}{2})(T-t) }{\sigma\sqrt{T-t}} $$ Since $S_t$ is modelled as geometric brownian motion, it has to be continuous in $t$. I see that everything in the B&S formula is continuous in $t$. But I can not proof the continuity $$ C_t \rightarrow C_T \hspace{1cm} \text{when } t\rightarrow T $$