Let U CRX RN be open and let (to, xo) U. Choose a, r >...

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Let U CRX RN be open and let (to, xo) € U. Choose a, r > 0 such that Q [toa, to +a] × B(xo, r) ≤ U. Suppose that f € C(U, RN). Set M := sup(t₁x) €Q |ƒ (t, x) |∞ <∞ and b := to + min{a, r/M}. (a) Briefly explain why, given & > 0, there exists 6 > 0 such that \f(t, x) = f(s, y) |∞ < & for all (s,x), (t, y) = Q with max{[t − s], \x − y|∞} <d. (b) Given & > 0 choose 6 > 0 as in part (a). Let to < t₁ < ··· < tm = b be a partition P of [to, b] with |||P|| := maxk=1,...,m(tk – tk-1) ≤ min{6, 8/M}. Let uo := xo and inductively define uk := Uk-1 + f(tk-1, Uk-1) (tk - tk-1) where k = 1, ..., m. Define a piecewise constant function g: [to, b] → RN by g(to) = f(to, xo) and g(T) = f(tk-1, Uk-1) if T € (tk-1, tk], where k = 1,. .. ,m. Then let u(t) := xo + + [ 8 (7) dr. to Show that u (tk)= uk for k = 0,..., m - 1 and that for all s,te [to, b]. |u(t)u(s) ∞ ≤ M t - s| Let U CRX RN be open and let (to, xo) € U. Choose a, r > 0 such that Q [toa, to +a] × B(xo, r) ≤ U. Suppose that f € C(U, RN). Set M := sup(t₁x) €Q |ƒ (t, x) |∞ <∞ and b := to + min{a, r/M}. (a) Briefly explain why, given & > 0, there exists 6 > 0 such that \f(t, x) = f(s, y) |∞ < & for all (s,x), (t, y) = Q with max{[t − s], \x − y|∞} <d. (b) Given & > 0 choose 6 > 0 as in part (a). Let to < t₁ < ··· < tm = b be a partition P of [to, b] with |||P|| := maxk=1,...,m(tk – tk-1) ≤ min{6, 8/M}. Let uo := xo and inductively define uk := Uk-1 + f(tk-1, Uk-1) (tk - tk-1) where k = 1, ..., m. Define a piecewise constant function g: [to, b] → RN by g(to) = f(to, xo) and g(T) = f(tk-1, Uk-1) if T € (tk-1, tk], where k = 1,. .. ,m. Then let u(t) := xo + + [ 8 (7) dr. to Show that u (tk)= uk for k = 0,..., m - 1 and that for all s,te [to, b]. |u(t)u(s) ∞ ≤ M t - s|