Question: Let $\phi(t)$ be a positive continuous function on $[0, \infty)$ and $f(t,x)$ be continuous function of two variables so that $|f(t,x)| \leq \phi(t) |x|$. Suppose $\int_0^\infty \phi(t) < \infty.$ Show that if the function $y$ satisfies the inequality $$|y(t)| \leq \int_0^t |f(s, y(s))| \, ds,$$ for all $t \in [0, \infty)$. Then, $y(t) \equiv 0$.
My attempt: If $y$ is a measurable function defined on $[0, \infty)$, let $\mu$ be a finite positive measure defined on $[0, \infty)$ so that $\mu(E) := \int_E \phi $ for every measurable $E \subset [0, \infty)$. Then, by hypothesis we have the following inequality $$|y(t)| \leq \int_0^t |y(s)|\, d \mu(s)$$ However, I am stuck as I'm not sure how to use the fact that $f$ is continuous function. Any hints/solutions are appreciated.