Exercise :
Find all the continuous and non-negative functions $f(t)$ such that:
$$f(t) \leq \int_0^t f(s)ds$$
Attempt :
I know there are some more "brutal" solving ways to this, such as a Laplace transformation or probably Mean Value Theorem tricks, but my idea is using the Gronwall Inequality, which says :
Let $φ(t)$ be a continuous function in $[0,T]$. Suppose that $\exists k>0$ and another continuous function $f(t)$, such that :
$$ φ(t) \leq f(t) + k\int_o^tφ(u)du, \forall t \in [0,T]$$
Then : $$φ(t) \leq f(t) + k\int_0^t f(τ)e^{t-τ}dτ,\forall t \in [0,T]$$
My problem is that I cannot see how to properly use the inequality or how to fix it, so I can show what is asked.
Any help would be appreciated !