Which has one continuous partial derivative with respect to

Question

Question: Show that a solution of the heat equation

∂u/∂t = K(∂²u/∂x²),

u(0, t) = u(L, t) = 0,

Which has one continuous partial derivative with respect to time, and two continuous partial derivatives with respect to space has the property that

Use this to conclude that solutions with this degree of smoothness are unique.