Differentiability implies ___________________
continuity
Differentiability implies the existence of the derivative of a function at a particular point. Another way of stating this is that a function f(x) is differentiable at a point if the limit of the difference quotient of f(x) as x approaches that point exists. In other words, if a function has a derivative at a particular point, then it is smooth and has a well-defined slope at that point. Therefore, differentiability also implies continuity of the function at that point, as a function cannot be differentiable at a point where it is not continuous.
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