Does a differential equation always have continuous solutions? The one-dimensional time-independent Schroedinger Equation is defined as
$$E\phi(x)=-\frac{\hbar^2}{2m}\frac{d^2\phi(x)}{dx^2}+V(x)\phi(x)$$
In my QM1 lecture, the Professor said that the solutions $\phi(x)$ are continuous because the Schroedinger Equation is a differential Equation. Why is this the case? Which theorem states that the solutions to differential equations are continuous?

Chicken or egg?
It is the other way round: one writes differential equations for the quantities described by continuous functions. At the discontinuities the derivative of the functions (at least some of them) are poorly defined, which may render the equation involving these derivatives meaningless. Thus, Schrödinger equation implies that the wave functions are at least twice differentiable.
Everyday life (of a physicist)
In practice one does work with discontinuous functions, which is either a bit of cheating or an approximation grounded in the more sophisticated theory of generalized functions. The most-encountered example is the delta-function potential, where the derivative of the wave function experiences a jump. This could be justified using the theory of generalized functions or by considering a very narrow rectangular well/barrier. (Note that the discontinuity of the potential does not cause trouble, as it is not differentiated - it simply translates into the boundary conditions.)
