exercises_10 - and g are continuous. Sketch the graphs of f...

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Analysis 1: Exercises 10 1. (a) Let f be continuous at a R . Prove that | f | is continuous at a . (b) Let f and g be continuous at a R . Prove that max { f,g } and min { f,g } are continuous at a . 2. Let f and g be continuous at a R . Prove that (a) f + g is continuous at a . (b) f · g is continuous at a . If, in addition, g ( a ) 6 = 0, then (c) f /g is continuous at a . 3. Let f : R R . Suppose that ( M > 0)( x R ) h | f ( x ) | ≤ M | x | i . Prove that f is continuous at 0. 4. Let f : R R be continuous at a and suppose that f ( a ) > 0. Show that there exists a δ -neighbourhood B ( a,δ ) = ( a - δ,a + δ ) of a such that whenever x B ( a,δ ) then f ( x ) > 0. 5. Investigate continuity of the following functions (a) f ( x ) = ± 2 x if 0 x 1 , 2 - x if 1 < x 2 . (b) f ( x ) = ± x 2 if 0 x 1 , 2 - x if 1 < x 2 . (c) f ( x ) = ± x 2 - 4 x - 2 if x 6 = 2 , A if x = 2 . (d) f ( x ) = lim n →∞ x n 1 + x n ( x 0). 6. Let f ( x ) = [ x ] be the greatest integer that is less than or equal to x , and let g ( x ) = x - [ x ]. Determine the points on R at which f
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Unformatted text preview: and g are continuous. Sketch the graphs of f and g . 7. Let f : R R be Dirichlets function dened by f ( x ) = 1 if x is rational , 0 if x is irrational . Prove that f is discontinuous at every point x R . 8. Let f : (0 , 1) R be dened as f ( x ) = if x is irrational , 1 q if x = p q in lowest terms . 2 (Recall that p q is in lowest terms if p and q are integers with no common factor and q &amp;gt; 0.) Prove that f is continuous at every irrational x and discontinuous at every rational x . 9. Give an example of a function dened on R which is continuous at only one point and discontinuous at all the other points on R ....
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exercises_10 - and g are continuous. Sketch the graphs of f...

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