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Math Help - Continuity and Differentiability

  1. #1
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    Continuity and Differentiability

    Write about the continuity and differentiability of each function at the indicated point of its domain. If you claim a function is not continuous at the given point, support the claim with appropriate documentation based on the continuity test. If claiming non-differentiability, support that claim.

    f(x)=(x-1)^(2/3); x=1

    g(x)= abs(x-2) ; x=2
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by warriors837 View Post
    Write about the continuity and differentiability of each function at the indicated point of its domain. If you claim a function is not continuous at the given point, support the claim with appropriate documentation based on the continuity test. If claiming non-differentiability, support that claim.

    f(x)=(x-1)^(2/3); x=1

    g(x)= abs(x-2) ; x=2

    Differentiability implies continuity, hence if a function is not continuous at a point, it is not differentiable either.

    f(x) is continuous at x=1 because \lim_{x \to 1} f(x) = f(1). This can be verified numerically, and by graphing.

    f(x) will fail to be differentiable if \lim_{x \to 1}\frac{f(x)-f(1)}{x-1} does not exist.

    You can numerically investigate the limit from the left, and the limit from the right. If the two match, then the function is differentiable. If they don't, it fails.
    Graphically, if you have a corner or a cusp (since it is continuous) then it is not differentiable.

    Follow a similar logic in the second problem.

    Good luck!
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