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Math Help - Prove a closed set interval is continuous

  1. #1
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    Prove a closed set interval is continuous

    Let a be less than b which is less than c.
    Suppose that f is continuous on [a,b], that g is continuous on [b,c], and that f(b)=g(b).
    Define h on [a,c] by h(x):= f(x) for x in [a,b] and h(x):= g(x) for x in (b,c].
    Prove that h is continuous on [a,c]
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by gixxer998 View Post
    Let a be less than b which is less than c.
    Suppose that f is continuous on [a,b], that g is continuous on [b,c], and that f(b)=g(b).
    Define h on [a,c] by h(x):= f(x) for x in [a,b] and h(x):= g(x) for x in (b,c].
    Prove that h is continuous on [a,c]
    The only point in question is b. Since f is continuous, \lim_{x\to b^-}h(x)=\lim_{x\to b^-}f(x)=f(b)=h(b)

    Since g is continuous, \lim_{x\to b^+}h(x)=\lim_{x\to b^+}g(x)=g(b)=f(b)=h(b)

    So \lim_{x\to b^-}h(x)=h(b)=\lim_{x\to b^+}h(x), and h is continuous at b and therefore on [a,c].
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