Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps
$$
0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0.
$$
We can take inductive limit (as TVS) and get:
$$
0\to B\;{\xrightarrow {\ f\ }}\;C\;{\xrightarrow {\ g\ }}\;D\to 0.
$$
I know, that direct limit preserves exactness. But I would like to know, new maps will remain continuous?

Thank you so much!