Forward Euler is convergent under mild conditions on $f(t,x)$, as explained below

Let $\delta t$ be the time step size parameter (assumed to be constant for clarity's sake), let $T$ be the time span of simulation and set $t_k = k \delta t$ for any $k \in \mathbb{N}_0$.  By integration by parts: 
$$
y(t_{k+1}) = y(t_k) + f(t_k,y(t_k)) + \int_{t_k}^{t_{k+1}} (t_{k+1} - s) \frac{d}{ds} f(s, y(s)) ds 
$$
Let $\epsilon_k$ denote the global error of the forward Euler scheme.  Since $f$ is Lipschitz we have that: 
$$
\epsilon_{k+1} \le (1+ L_f \delta t) \epsilon_k + \| \int_{t_k}^{t_{k+1}} (t_{k+1} - s) \frac{d}{ds} f(s, y(s)) ds \|
$$
where $L_f$ is the Lipschitz constant of $f$.
By Cauchy-Schwarz inequality: 
$$
\epsilon_{k+1} \le (1+ L_f \delta t) \epsilon_k + (\int_{t_k}^{t_{k+1}} |(t_{k+1} - s)|^2 ds)^{1/2} (\int_{t_k}^{t_{k+1}} \|\frac{d}{ds} f(s, y(s))\|^2 ds )^{1/2} 
$$
which simplifies to:
$$
\epsilon_{k+1} \le (1+ L_f \delta t) \epsilon_k + \delta t^{3/2} M 
$$
where $M$ satisfies
$$
(\int_{0}^{T} \|\frac{d}{ds} f(s, y(s))\|^2 ds )^{1/2} \le M \tag{$\star$}
$$
By induction (or discrete Gronwall's Lemma), it follows that:
$$
\epsilon_{k} \le \frac{e^{L_f T}  M}{L_f} \; \delta t^{1/2}
$$
for all $k \in \mathbb{N}_0$ satisfying $t_k < T$.  Note that ($\star$) may hold even when $f(t,x)$ is just Lipschitz-continuous in $x$. Indeed, [Rademacher's Theorem][1] guarantees that a Lipschitz function is differentiable almost everywhere.  The price for this more mild assumption is that the theoretical rate of convergence drops from $\mathcal{O}(\delta t)$ to $\mathcal{O}(\delta t^{1/2})$. 

However, numerical evidence seems to indicate this estimate is a bit pessimistic.  Consider the initial value problem 
$$
\dot y=|y|/2-(y-1) \;, \quad y(0)=-2.3 \;,
$$
where the right hand side is Lipschitz, but not differentiable at $0$. Here is a figure of the true solution over the time interval $[0,1]$.  (I selected this solution so that $y(1)$ is close to the point where the right hand side of the differential equation $|x|/2-(x-1)$ is not differentiable.)

![true solution][2]

Here is a figure of the relative error of forward Euler with respect to a converged target.  (This metric of convergence is commonly used in the absence of an analytical solution.)

![relative error][3]

For a MATLAB function file that reproduces this last figure [click here][4].  


  [1]: http://en.wikipedia.org/wiki/Rademacher%27s_theorem
  [2]: https://i.sstatic.net/5pxjp.jpg
  [3]: https://i.sstatic.net/fJhCd.jpg
  [4]: http://www.crab.rutgers.edu/~nb361/mysoftware/lipschitz.m