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00fall-minitest2

00fall-minitest2 - x(iv If f g IR 7→ IR are continuous at...

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MAT 371 Advanced Calculus / 50 Oct 12, 2000 Minitest 2 name 1. State the standard definition for continuity of a function at a point. (10 pts) 2. Using basically only the ε - δ -characterization of continuity at a point ( “the standard definition ”) prove the following: (30 pts) (i) Every constant function is continuous at every point. (ii) The identity function f : x x is continuous at every point. (iii) If f, g : IR IR are continuous at x 0 IR then f + g is continuous at
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Unformatted text preview: x . (iv) If f, g : IR 7→ IR are continuous at x ∈ IR then f · g is continuous at x . 3. Explain how the results of 2. imply as a corollary that every polynomial function p : IR 7→ IR is continuous at every point x ∈ IR. (10 pts) (No detailed proof expected, just a general outline of the argument.) Bonus : Under the assumptions of 2.iii, show that f g is also continuous at x provided g ( x ) 6 = 0....
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