If f is a real-valued differentiable function satisfying |f(x)-f(y)| ≤(x-y)^2, x, y ∈R and f(0)=

step 1 We are given that function f is differentiable which means we can write that f dash of x is equa

If f is a real-valued differentiable function satisfying...

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Key Concepts

Quadratic Bound on Function Differences

The inequality |f(x) - f(y)| ? (x - y)² imposes a quadratic restriction on how much the function can change between any two points. This strong constraint implies that as the difference between x and y becomes small, the change in f becomes negligibly small, ultimately forcing the derivative of f to be zero and the function to be constant.

Constant Function

If the derivative of a function is zero everywhere, the function must be constant. This is a direct consequence of the Mean Value Theorem combined with the given bound on the difference of function values, leading to the conclusion that the function does not change at all.

Mean Value Theorem

The Mean Value Theorem states that for a function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the instantaneous rate of change (the derivative) equals the average rate of change over that interval. This concept helps in deducing that if the difference between function values grows much slower than linearly, then the derivative must be zero.

Differentiability

Differentiability means that the function has a derivative at every point in its domain. This property allows one to apply the Mean Value Theorem, which is essential for analyzing how the function behaves between any two points.

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