f07HW1 - ( a, b ). ii) Suppose functions f, g : ( a, b ) R...

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MAT 125A Homework 1 M.Fukuda October 2, 2007 Please submit your answers at the discussion session on October 11th Thursday. 1. Let f ( x ) = 1 - x x < 0 0 x = 0 1 x x > 0 . i) Find at which point of R the above f is continuous or discontinuous. ii) Prove your idea. 2. i) Show that f ( x ) = x 3 is continuous on R by using ǫ - δ property of Theorem 17.2. ii) Show that g ( x ) = x 3 x 6 + 1 is continuous on R . 3. i) Suppose a function f : ( a, b ) R is continuous, and f ( r ) = 0 for each rational number r in ( a, b ). Show that f ( x ) = 0 for each x
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Unformatted text preview: ( a, b ). ii) Suppose functions f, g : ( a, b ) R are continuous, and f ( r ) = g ( r ) for each rational number r in ( a, b ). Show that f ( x ) = g ( x ) for each x ( a, b ). 4. i) If we replace the closed interval [ a, b ] in theorem 18.1 by the open interval ( a, b ) the proof doesnt work. Explain why. ii) Show that the function in the question 1 has neither the maximum value or the minimum value on (1 , 2). 1...
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