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Math Help - continuous, not differentiable

  1. #1
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    continuous, not differentiable

    Given the function:

    <br /> <br />
f(x) = a |x - k| + b<br />

    where a, b and k are constants, show that f(x) is continuous but not differentiable at x = k

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  2. #2
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    continuous at x = k ...

    f(k) = b ... f(k) exists

    \lim_{x \to k^-} f(x) = \lim_{x \to k^+} f(x) = b = f(k)

    therefore, f(x) is continuous at x = k



    differentiability at x = k ...

    \lim_{x \to k} \frac{f(x) - f(k)}{x - k} =

    \lim_{x \to k} \frac{a|x-k|+b - b}{x - k} =

    \lim_{x \to k} \frac{a|x-k|}{x - k} =

    a if x \to k^+

    -a if x \to k^-

    since the limit does not exist, f(x) is not differentiable at x = k.
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  3. #3
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    thanks skeeter!
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