If |f(w)-f(x)| ≤|w-x| for all values w and x and f is a differentiable function, show that -1 ≤f

step 1 Okay, so we're assuming that for any numbers W and X, we have F of W minus F of X, is less than

If |f(w)-f(x)| ≤|w-x| for all values w and x and f is a...

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Key Concepts

Lipschitz Continuity

Lipschitz continuity is a condition that restricts the rate at which a function can change. A function that satisfies the inequality |f(w) - f(x)| ? |w - x| for all inputs is said to be Lipschitz continuous with a Lipschitz constant of 1. This condition implies that the function does not oscillate or grow too quickly, establishing a uniform bound on its rate of change.

Definition of the Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the interval approaches zero. Given the Lipschitz condition, the absolute value of the difference quotient |(f(w) - f(x))/(w - x)| is bounded by 1 for all non-zero intervals. Therefore, in the limit where the interval goes to zero, the derivative, which represents the instantaneous rate of change, must also satisfy |f?(x)| ? 1.

Mean Value Theorem

The Mean Value Theorem states that for a continuous function on a closed interval and differentiable on the open interval, there exists a point where the derivative equals the average rate of change over that interval. This theorem connects the global behavior of the function, as represented by the Lipschitz condition, with the local behavior of its derivative, reinforcing the idea that the derivative cannot exceed the Lipschitz constant in absolute value.

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