Explain where the fact that a differentiable function must be continuous was assumed in the proo

step 1 So for the following problem, we want to explain where the fact that a differential function mus

Explain where the fact that a differentiable function must...

Key Concepts

Differentiability Implies Continuity

A key concept in this context is that if a function is differentiable at a certain point, it must also be continuous at that point. This is crucial because in the process of proving rules such as the product rule, continuity is used to transition from difference quotients to the limit expressions that define the derivative. Without the guarantee of continuity, the limit manipulations required in the proof would not hold.

Product Rule for Derivatives

The product rule is a fundamental result in calculus, stating that the derivative of a product of two functions is given by the derivative of the first function times the second function plus the first function times the derivative of the second. In the proof of this rule, the notion that differentiable functions are also continuous is implicitly used when evaluating the limit involved in the definition of the derivative.

Limit Definition of the Derivative

This concept involves defining the derivative as the limit of the difference quotient as the interval approaches zero. In proofs such as that of the product rule, the limit definition is employed to split the expression into manageable parts. The continuity of differentiable functions ensures that limit operations, such as substituting the function’s value at a point, are valid and can be seamlessly incorporated into the proof.

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