It is proved that if f (x) is differentiable on the interval I and the derivative function on I is bounded, then f (x) is uniformly continuous on I

Let f '(x) ｜ ≤ M
Then, for any x, y ∈ I, according to Lagrange's mean value theorem, f (y) - f (x) ≤ m y-x
Therefore, for any given ε > 0, take δ = ε / m, then when Y-X ﹤ δ, ﹤ f (y) - f (x) ﹤ m ﹤ Y-X ﹤ m (ε / M) = ε
The proposition is proved, and the proof is finished

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