RealAnalysis Math 125A, Fall 2012 Sample Final Questions

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Real Analysis math 125A, fall 2012 . Sample Final Questions 1. Define f : R R by x3. f (x) = . 1 + x2. Show that f is continuous on R. Is f uniformly continuous on R? Solution. To simplify the inequalities a bit, we write x3 x 2. =x . 1+x 1 + x2. For x, y R, we have x y |f (x) f (y)| = x y . +. 1 + x2 1 + y 2 x y |x y| + 2. 2. . 1+x 1+y Using the inequality 2|xy| x2 + y 2, we get y x y + xy 2 x2 y x 1 + x2 1 + y 2 = (1 + x2 )(1 + y 2) . 1 + |xy|. |x y|. (1 + x2 )(1 + y 2 ). 1 1 + x2 + 1 + y 2. . |x y|. 2 (1 + x2 )(1 + y 2 ). . 1 1 1. + |x y|. 2 1 + y 2 1 + x2. |x y|. It follows that |f (x) f (y)| 2|x y| for all x, y R.

RealAnalysis Math 125A, Fall 2012 Sample Final Questions 1. Define f : R→ Rby f(x) = x3 1+x2 Show that f is continuous on R. Is f uniformly continuous on R?

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