differentiable function

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The real function f of one variable is differentiable at a if (f (a+h)−f (a)/h has a limit as h → 0; that is, if the derivative of f at a exists. The rough idea is that a function is differentiable if it is possible to define the gradient of the graph y=f (x) and hence define a tangent at the point. The function f is differentiable in an open interval if it is differentiable at every point in the interval; and f is differentiable on the closed interval [a, b], where a < b, if it is differentiable in (a, b) and if the right derivative of f at a and the left derivative of f at b exist.

Subjects: Mathematics.

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