for the irrational number ln 2 ≈ 0 . 6931. This series was known and used by Euler. Although we can integrate uniformly convergent sequences, we cannot in gen- eral differentiate them. In fact, it’s often easier to prove results about the conver- gence of derivatives by using results about the convergence of integrals, together with the fundamental theorem of calculus. The following theorem provides suffi- cient conditions for f n → f to imply that f ′ n → f ′ . Theorem 1.64. Let f n : ( a, b ) → R be a sequence of differentiable functions whose derivatives f ′ n : ( a, b ) → R are integrable on ( a, b ). Suppose that f n → f pointwise and f ′ n → g uniformly on ( a, b ) as n → ∞ , where g : ( a, b ) → R is continuous. Then f : ( a, b ) → R is continuously differentiable on ( a, b ) and f ′ = g . Proof. Choose some point a < c < b . Since f ′ n is integrable, the fundamental theorem of calculus, Theorem 1.45, implies that f n ( x ) = f n ( c ) + integraldisplay x c f ′ n for a < x < b. Since f n → f pointwise and f ′ n → g uniformly on [ a, x ], we find that f ( x ) = f ( c ) + integraldisplay x c g. Since g is continuous, the other direction of the fundamental theorem, Theo- rem 1.50, implies that f is differentiable in ( a, b ) and f ′ = g . square In particular, this theorem shows that the limit of a uniformly convergent se- quence of continuously differentiable functions whose derivatives converge uniformly is also continuously differentiable. The key assumption in Theorem 1.64 is that the derivatives f ′ n converge uni- formly, not just pointwise; the result is false if we only assume pointwise convergence of the f ′ n . In the proof of the theorem, we only use the assumption that f n ( x ) con- verges at a single point x = c . This assumption together with the assumption that f ′ n → g uniformly implies that f n → f pointwise (and, in fact, uniformly) where f ( x ) = lim n →∞ f n ( c ) + integraldisplay x c g. Thus, the theorem remains true if we replace the assumption that f n → f pointwise on ( a, b ) by the weaker assumption that lim n →∞ f n ( c ) exists for some c ∈ ( a, b ). This isn’t an important change, however, because the restrictive assumption in the theorem is the uniform convergence of the derivatives f ′ n , not the pointwise (or uniform) convergence of the functions f n . The assumption that g = lim f ′ n is continuous is needed to show the differ- entiability of f by the fundamental theorem, but the result result true even if g isn’t continuous. In that case, however, a different (and more complicated) proof is required.
40 1. The Riemann Integral 1.9.2. Pointwise convergence. On its own, the pointwise convergence of func- tions is never sufficient to imply convergence of their integrals. Example 1.65. For n ∈ N , define f n : [0 , 1] → R by f n ( x ) = braceleftBigg n if 0 < x < 1 /n, 0 if x = 0 or 1 /n ≤ x ≤ 1 . Then f n → 0 pointwise on [0 , 1] but integraldisplay 1 0 f n = 1 for every n ∈ N . By slightly modifying these functions to f n ( x ) = braceleftBigg n 2 if 0 < x < 1 /n, 0 if x = 0 or 1 /n ≤ x ≤ 1 , we get a sequence that converges pointwise to 0 but whose integrals diverge to ∞ .
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