It is know that a sufficient and necessary condition for $$\dot y(t) = f(y(t), t), \quad t > 0, \quad y(0) = y_0$$ assumes a unique solution of $f$ is Lipschitz in $y$ and continuous in $t$. However, I didn't find in the literature that this condition could guarantee the convergence of Euler's scheme (forward or backward) to the solution. Instead, additional condition imposed on $y$ that $y$ is $C^2$ is seems needed, for example http://persson.berkeley.edu/228A/Fall10/doc/lec05-2x3.pdf. I was wondering that was it possible to weaken the condition for the convergence of Euler's scheme?

It is known that a sufficient and necessary condition for $$\dot y(t) = f(y(t), t), \quad t > 0, \quad y(0) = y_0$$ to have a unique solution is $f$ Lipschitz in $y$ and continuous in $t$. However, I didn't find in the literature that this condition could guarantee the convergence of Euler's scheme (forward or backward) to the solution. Instead, additional condition imposed on $y$ that $y$ is $C^2$ seems needed, see for example http://persson.berkeley.edu/228A/Fall10/doc/lec05-2x3.pdf. I was wondering whether it was possible to weaken the condition for convergence of Euler's scheme.