PDE Initial Condition Question

Hello,

I have the following PDE:

$\displaystyle u_t = k u_xx$

with $\displaystyle u(x,0) = f(x), u_x(0,t) = u_x(L,t) = 0$

which I solve using separation of variables by letting $\displaystyle u = XT$.

Now, for the conditions we have

$\displaystyle u_x(0,t) = X'(0) T(t) = 0 $

so $\displaystyle X'(0) = 0$ and

$\displaystyle u_x(L,t) = X'(L) T(t) = 0 $

$\displaystyle X'(L) = 0$

My question is why can't $\displaystyle T(t) = 0$ in these cases? This assumption is made in every PDE I've solved using sep of vars, but I would like to know why exactly this is.

Thanks