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Exercise_Answers-Continuity - Continuity and Limits of...

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Continuity and Limits of Functions Exercise Answers 1. Let f be given by f ( x ) = 4 - x for x 4 and let g be given by g ( x ) = x 2 for all x R . (a) dom( f + g ) = dom( fg ) = ( -∞ , 4], dom( f g ) = [ - 2 , 2] and dom( g f ) = ( -∞ , 4] (b) ( f g )(0) = 2, ( g f )(0) = 4, ( f g )(1) = 3,( g f )(1) = 3, ( f g )(2) = 0 and ( g f )(2) = 2. (c) No! (d) ( f g )(3) is not, but ( g f )(3) is. 2. Let f be given by f ( x ) = 4 for x 0, f ( x ) = 0 for x < 0 and let g be given by g ( x ) = x 2 for all x R . (a) f + g : R R is given by ( f + g )( x ) = 4 + x 2 for all x 0 and ( f + g )( x ) = x 2 for all x < 0. fg : R R is given by ( fg )( x ) = 4 x 2 for all x 0 and ( fg )( x ) = 0 for all x < 0. f g : R R is given by ( f g )( x ) = 4 for all x R . g f : R R is given by ( g f )( x ) = 16 for all x 0 and ( g f )( x ) = 0 for all x < 0. (b) The functions g , fg and f g , are continuous, while f , f + g and g f are discontinuous because they are discontinuous at 0. Note that although f is discontinous, the function fg is continuous – so the continuity of fg does not imply both f and g are continuous. 3. The functions given by sin x , cos x , e x , 2 x , ln x for x > 0, and x p for x > 0 ( p R ) are continuous on their domains. Use these facts and theorems in the notes to prove that the functions given as below are also continuous. 1 (a) ln(1 + cos 4 x ). Solution. First 1 = x 0 is continuous by one of the above facts. Next, cos x is continuous from above, and cos 4 x = (cos x ) 4 is a composition of two functions (given by cos x and y 4 ) which are continuous on their domains. Thus, since the composition of two continuous functions is continuous, cos 4 x is continuous. Now, since a function formed by pointwise addition is continuous, it follows that (1 + cos 4 x ) is a continuous function. Finally, since cos 4 x 0 for all x R , we have 1 + cos 4 x 1 > 0 for all x R . Thus since ln x is continuous for x > 0, the composition on the right of ln with (1+cos 4 x ) is continuous. 1 We have used cos n x to denote (cos x ) n for any n N and similarly for sin n x . 1

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(b) [sin 2 x + cos 6 x ] π (c) 2 x 2 (d) 8 x (e) tan x for x 6 = odd multiple of π/ 2 (f) x 2 sin(1 /x ) for x 6 = 0 (g) x 2 sin(1 /x ) for x 6 = 0 (h) (1 /x ) sin(1 /x 2 ) for x 6 = 0 4. Prove that the function x is continuous on its domain [0 , ). Hint : use the se- quential definition of continuity and the fact that if ( s n ) is a sequence of nonnegative real numbers and s = lim s n , then lim s n = s . 5. (a) Prove that if m N , then the function f ( x ) = x m is continuous on R . Hint : You can construct an ε - δ proof using the identity x m - y m = ( x - y )( x m - 1 + x m - 2 y + · · · + xy m - 2 + y m - 1 ) . Or you can prove the result using induction on m . First prove that g ( x ) = x is continuous on R . Then use an inductive hypothesis that f ( x ) = x m is continuous on R and the theorem about the continuity of ( fg ). Solution. Let ε > 0 and let x 0 be an arbitrary real number. We need to find a δ > 0 such that if | x - x 0 | < δ then | x m - x m 0 | < ε . Now, by the hint, | x m - x m 0 | = | x - y || x m - 1 + x m - 2 x 0 + · · · + xx m - 2 0 + x m - 1 0 | .
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Exercise_Answers-Continuity - Continuity and Limits of...

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