结论错误.
反例如f(x) = 3x^(4/3)·sin(1/x)(补充定义f(0) = 0).
可知f'(0) = lim{x → 0} f(x)/x = lim{x → 0} 3x^(1/3)·sin(1/x) = 0.
而在x ≠ 0处, f'(x) = 4x^(1/3)·sin(1/x)-3x^(-2/3)·cos(1/x), 故f(x)在[-1,1]上可导.
当k为正整数, f'(1/(2kπ)) = -3(2kπ)^(2/3).
k → ∞时, 1/(2kπ) → 0, 但f'(1/(2kπ)) → -∞, 故f'(x)在[-1,1]上无界.
反例如f(x) = 3x^(4/3)·sin(1/x)(补充定义f(0) = 0).
可知f'(0) = lim{x → 0} f(x)/x = lim{x → 0} 3x^(1/3)·sin(1/x) = 0.
而在x ≠ 0处, f'(x) = 4x^(1/3)·sin(1/x)-3x^(-2/3)·cos(1/x), 故f(x)在[-1,1]上可导.
当k为正整数, f'(1/(2kπ)) = -3(2kπ)^(2/3).
k → ∞时, 1/(2kπ) → 0, 但f'(1/(2kπ)) → -∞, 故f'(x)在[-1,1]上无界.
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