Metropolis-Hastings undersampling near kink in distribution function

Question

I'm trying to use Metropolis-Hastings to sample from a distribution that's very close to

$$\exp(-|x|/\ell)$$

and I'm finding that the method is undersampling near the origin, where there's a kink in the distribution function. I've attached an example figure. Is there a smoothness requirement to Metropolis-Hastings that isn't discussed that often? If so, is there a good workaround for this or alternative algorithm?

Edit: Here is a minimal working example, along with the output.

def exp_func(x, ell):

    return np.exp(-np.abs(x)/ell)

ell = 2.

sigma = 1.

n_samples = 5000
sequence = np.zeros(n_samples)

theta_t = np.random.random()
idx = 0

while idx < n_samples:
    theta_star = np.random.normal(loc=theta_t, scale=sigma)
    alpha = min(exp_func(theta_star, ell)/exp_func(theta_t, ell), 1.)
    u = np.random.random()
    if u < alpha:
        theta_t = theta_star
        sequence[idx] = theta_star
        idx += 1

and what it produces:

Answer 1

Differentiability or even continuity is not a requirement for Metropolis to apply (and converge) and the fit is quite good except for the very centre, so I rather foresee a coding error.

Here is an illustration of plain Metropolis working out very well on an example from our book for the target function$$f(x) \propto \exp(-x^2/2) (\sin(6x)^2+3\cos(x)^2\sin(4x)^2+1)$$which gets easily simulated by accept-reject (Section 2.3.1).

x=c()
for(t in 1:1e4){
  y=rnorm(1)#standard Normal proposal
  x=c(x,ifelse(runif(1)<f(y)*dnorm(z)/f(z)/dnorm(y),z<-y,z))}

with indeed a very good fit:

If one forgets to duplicate the current value under a rejection, i.e., to set $X_{t+1}=X_t$, as in

 for(t in 1:1e4){y=rnorm(1)
   if(runif(1)<f(y)*dnorm(z)/f(z)/dnorm(y))x=c(x,z<-y)}

the fit is lost:

For the Laplace distribution, here is a comparison between replicating and not-replicating: