Basic Analysis I — Real Analysis · Problem 7.6.2

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · fixed-point-theorem

Problem

In the proof of Picard's theorem, show that if f:[−h,h]→Jf: [-h,h] \to J is continuous, then F(t,f(t))F\bigl(t, f(t)\bigr) is continuous on [−h,h][-h,h] as a function of tt. Use this to show that T(f)(x)≔y0+∫0xF(t,f(t)) dtT(f)(x) \coloneqq y_0 + \int_0^x F\bigl(t, f(t)\bigr)\,dt is well-defined and that T(f)∈C([−h,h],R)T(f) \in C\bigl([-h,h],\mathbb{R}\bigr).
Intuitive explanation

The key insight is that composition of continuous functions stays continuous, and integration preserves continuity. Since ff is continuous on [−h,h][-h,h] and FF is continuous on the rectangle [−h,h]×J[-h,h] \times J, the composite function t↦F(t,f(t))t \mapsto F(t, f(t)) inherits continuity by the composition rule. Once you know the integrand is continuous, the fundamental theorem of calculus guarantees that T(f)T(f) — defined as a definite integral with continuous integrand — is itself a continuous function on [−h,h][-h,h]. This is how Picard's iteration operator maps the space of continuous functions back into itself.

Intuitive explanation

The key insight is that composition of continuous functions stays continuous, and integration preserves continuity. Since ff is continuous on [−h,h][-h,h] and FF is continuous on the rectangle [−h,h]×J[-h,h] \times J, the composite function t↦F(t,f(t))t \mapsto F(t, f(t)) inherits continuity by the composition rule. Once you know the integrand is continuous, the fundamental theorem of calculus guarantees that T(f)T(f) — defined as a definite integral with continuous integrand — is itself a continuous function on [−h,h][-h,h]. This is how Picard's iteration operator maps the space of continuous functions back into itself.

Related theory

Browse the problem library