For each natural number n and each number x in (-1, 1), define f₁(x)=√√√x² √ x² + = ₁ and define f(x) = |x|. Prove that the sequence (ƒ: (-1, 1)→ R} converges uni- formly to the function f: (-1, 1)→ R. Check that each function f: (-1, 1)→ Ris differentiable, whereas the limit function ƒ: (−1, 1) → R is hot differentiable. Does this contradict Theorem 9.19? Thm Let I be an open interval. Suppose that (f: I→ R) is a sequence of continuously differentiable functions that has the following two properties: 9.19. (i) The sequence {f: 1 → R} converges pointwise to the function f: 1 → R and (ii) The sequence of derivatives {f:I→ R} converges uniformly to the function 8:1 → R. Then the function f:I → R is continuously differentiable and f'(x) = g(x) for all x in [a, b].