Differentiation Requires More

Real Analysis · Axiom Academy

LESSON Differentiation Requires More Uniform convergence carries continuity across a limit — but not the derivative. To swap a limit and a derivative, the derivatives must converge uniformly. We already know one limit interchange works: if uniformly and each f_n is continuous, then f is continuous. Uniform convergence is exactly strong enough to carry continuity across the limit. So it is natural to push our luck and ask the same of the derivative. 2. The Counterexample: Uniformly Take the sequence of functions on : Claim: this converges uniformly to f(x) = 0 . The reason is the amplitude bound: The sine factor never exceeds 1 in size, so for every x at once we have . That bound is independent of x and tends to 0 , which is precisely uniform convergence. In the animation the whole wave is trapped inside a shrinking envelope that squeezes onto the axis as n climbs. 3. But the Derivatives Blow Up Differentiate term by term. The out front and the chain-rule factor of n combine into a : Watch what happened to the amplitude: it went from to . As n grows the derivative wave gets taller , not flatter. At any peak of the cosine, , which is unbounded in n . 4. Why Small Bumps Have Big Slopes The mechanism is geometric. The slope of a wave is set by both its height and how fast it wiggles. For f_n the height is but the frequency is n , and the chain rule multiplies them:

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