The Laplace method states that:
If $f(x)$ is twice differentiable on $[a,b]$ and $f(x)$ has a unique global maximum on $[a,b]$ at $x_0$ and $f''(x_0) <0$ \begin{align} \lim_{n \to \infty} \frac{ \int_a^b e^{n f(x)} dx}{ \sqrt{\frac{2 \pi}{- n f''(x_0)}} e^{n f(x0)}}=1. \end{align}
My question is does this method work if the function $f(x)$ is continuous but not differential at one only point on $[a,b]$?
For example, does Laplace method work \begin{align} f(x)=-|x|^3+x \end{align} for $[a,b]=[-1,1]$.
Is there a method that does this without splitting the integral into two parts $[-1,0)$ and $(0,1]$?
You can work with [-1,0] and [0,1] separately . In each of these two subintervals you can apply Laplace method.