About the existence of solution for linear occasion for first order equation

[Hyunmin Jung]

When p(t) is continuous on the interval containing initial t

but g(t) is not continuous on the I containing initial t, it violates theorem 2.4.1 but is unsure whether or

not non-unique solution exist for all t in I? and the case for when g(t) is continuous on the interval and p(t) is not.

[Victor Ivrii]

Linear equations tolerate a lot of abuse (non-smoothness). It does not follow from theorems we studied but there is an uniqueness and existence as long as $g$ is just integrable and $p$ is bounded