Mathematical methods for economic theory

Martin J. Osborne
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1.4 Exercises on intervals and functions

  1. At what points is the function
    f(x) = 
    left brace x2 − 2 if x ≤ 0
    −x2 if 0 < x < 1
    x2 − 2 if x ≥ 1
    continuous?

    Solution

    The function is continuous at all points except 0. We have f(0) = −2, but for x slightly greater than 0, f(x) is close to 0.

  2. Use the Intermediate Value Theorem to show that the equation x7 − 5x5 + x3 − 1 = 0 has a solution between −1 and 1.

    Solution

    Denoting the function by f, we have f(−1) = 2 > 0 and f(1) = −4 < 0. Thus by the Intermediate Value Theorem there exists x between −1 and 1 such that f(x) = 0.