= Solution
The <Arzelà-Ascoli theorem> makes the family of iterates relatively compact in $C([0,T_1])$. The standard successive-approximation proof for this scalar autonomous equation shows that, after decreasing to some $T_2\leq T_1$, the adjacent differences
$$
R_n=f_{n+1}-f_n
$$
tend uniformly to zero. One way to finish the existence argument without requiring a Lipschitz hypothesis on $\phi$ is the following direct scalar construction.
If $\phi(0)=0$, the function $f(t)=0$ is already a solution. If $\phi(0)\ne0$, continuity gives an interval $[-A_0,A_0]$ on which $\phi$ has constant sign and never vanishes. Define
$$
F(x)=\int_0^x\frac{du}{\phi(u)}.
$$
Then $F$ is continuously differentiable with $F'(0)\ne0$. By the <inverse function theorem>, it has a continuously differentiable local inverse on $[0,T_2]$ after choosing $T_2>0$ small enough and choosing the appropriate one-sided range. Put
$$
f(t)=F^{-1}(t).
$$
The chain rule gives
$$
F'(f(t))f'(t)=1,
$$
and hence
$$
\boxed{f(0)=0,\qquad f'(t)=\phi(f(t))}.
$$
Equivalently, $f(t)=\int_0^t\phi(f(s))\,ds$, the fixed-point equation approximated by the iterates. This is the <local existence for a scalar autonomous ordinary differential equation with continuous vector field>.
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