Given a seq, conclude ƒ(x) is diff'able at x = c?
Suppose that {Xn} is a sequence in an interval I that converges to the point c ∈ I. Suppose that the sequence
(ƒ(Xn) - ƒ(c) ) / ( Xn - c)
converges to a number L. Can we conclude that the function ƒ(x) is differentiable at x = c ?
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No.
If the sequence converges to c strictly from one side, then f(x) is only differentiable from that side -- nothing can be said about the convergence properties on the other side of x=c.
Let c to be the right-most point of the interval I that is still in I. Let {Xn} approach from the left, so that Xn-c is monotonically decreasing with n. The sequence probes the function entirely from the left, so you have a left-sided derivative, but no information about the function's behavior on the other side of c. For all you know, f(x) could be discontinuous at x=c+, making f(x) not differentiable at c since the limit from the left is L but the limit on the right could be different from L.
No it converges to C only on the one side. Just do the math lol