Sounds like a home work problem?

Note that 
$$g(T)=\lim_{t\to T-}g(t)=g(0)+\int\limits_0^T\tfrac{\partial}{\partial t}g$$
Then you get 
$$|D_0^m g(T)|\le \mathrm{Const}(m)$$
One can cover $M$ by charts with bounded $g(0)$-Christoffel symbols in each.
Then the above inequalities imply that $g(T)$ is $C^\infty$-smooth