# calculus

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For a function f(x) the derivative at x=2 is 5. Is the derivative of f(x) continuous at x=2? Thanks!

• calculus -

If a function has a derivative at a point, the function must be continuous there.

• calculus -

Isn't it asking if the derivative is continuous? Not whether the function is continuous?

• calculus -

Yes, I misread the question. My mistake.

According to this reference:
http://www.math.colostate.edu/~reinholz/ed/08fa_m160/lectures/mean_value_theorem.pdf
the existence of a derivative of a function at a point does not necessarily imply that the derivative is continuous. An example is provided of a function with derivatves throughout a region for which the derivative is not continuous at one point in the region.

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