Is continuity of the wavefunction "put in by hand" for the Dirac delta potentials?

Were $psi(x)$ not continuous at $x=0$ then $psi''(x)$ would contain the derivative of a $delta$-function, and there is nothing else in the equation $Hpsi=Epsi$ that could cancel it, so a discontinuity is not allowed.

1. Let us assume the integral form $$ psi(x)~=~ frac{2m}{hbar^2} int^{x}mathrm{d}y int^{y}mathrm{d}z (V(z)-E)psi(z) tag{1}$$ of the time independent 1D Schrödinger equation (TISE), where the potential $$V(x)~=~V_0delta(x)tag{2}$$ is a Dirac delta potential.

2. Let us assume that the wavefunction $psiin {cal L}_{rm loc}^1(mathbb{R})$ is a locally integrable function, so that the integral $int^{y}mathrm{d}z Epsi(z)$ in eq. (1) is well-defined.

3. Note in particular that we do not assume that $psi$ is continuous, cf. OP's title question. The following crucial question arises: How should we define the integral $$int^{y}mathrm{d}z V(z)psi(z)tag{3}$$ in eq. (1)?

   • Should we define (3) to be equal to $$V_0theta(y)psi(0)+C,tag{3'}$$ where $C$ is an integration constant?

   • Or if the left and right limits $psi(0^-)$ and $psi(0^+)$ exist, should we define (2) to be equal to $$V_0theta(y)frac{psi(0^-)+psi(0^+)}{2}+C~?tag{3''}$$

4. Whatever definition we propose for the Dirac delta distribution, it seems likely that (3) will become a locally integrable function of $y$.

   A mathematical bootstrap argument involving the TISE (1) then shows that $psiin C(mathbb{R})$ is continuous, cf. OP's title question. See also e.g. my related Phys.SE answer here.

You can think of the delta potential $V(x)=gdelta(x)$ as representing the limiting form of a potential barrier (or well) of height $Lambda$ and width $w$, with $wLambda=g$.

For any finite $Lambda$, $psi(x)$ and $psi’(x)$ must be continuous at both edges of the barrier. Within the barrier, Schrödinger’s equation implies that $psi’’(x)simLambda$, so the total change in $psi’(x)$ across the barrier is $sim g$.

This means that, in the limit $Lambdarightarrowinfty$ with $g$ fixed, $psi’(x)$ is finite everywhere within the barrier, but changes by a finite amount from one side of the barrier to the other. Since $psi’(x)$ is finite everywhere, $psi(x)$ is continuous.

The other answers are very good. I want to address one mis-think in the question.

"... if $psi$ is discontinuous at $x=0$, then we'll have trouble writing $psi(0)$ ..."

Not always. As a vast oversimplification, suppose that you have obtained $psi$ as a Fourier transform, which has $lim_{x rightarrow 0^-} psi(x) = -1$ and $lim_{x rightarrow 0^+} psi(x) = 1$. Then we know $psi(0) = 0$. At a step discontinuity, a Fourier series converges to the midpoint between the values to the left and right. (This is part of the Dirichlet conditions. There are mild technical hypotheses. The cited reference describes the real-valued version; the complex-valued version is essentially the same.)