. Suppose the function f(t, x) is piecewise continuous in t and locally Lipschitz in x uniformly across t in [t0, t1].

Problem 3. Suppose the function f (t, 3:) is piecewise continuous in t and locally Lipschitz in a: uniformly across t in [150,151]. Let W be a compact subset of R\". Show that SUPte[to,t1],mEW | f (t, 3:)| is nite. Hint: One common way to show a function is bounded everywhere is to x any point (s,y) in the domain. Noting that f (s,y) is nite, we can show the function itself is bounded if we can bound how far away f ($5,113) is from f (s,y) for any other (15,33). To start, it may be easier to consider the case where f (13,33) is continuous in t. Then, note that piecewise continuity states that the interval can be broken up into nitely many intervals where it is continuous, and the endpoints of the intervals agree with either the right or left limit