31 янв. 2013 г. · It is true that if f is defined on an interval in R and is everywhere differentiable with bounded derivative, then f is uniformly continuous. In ...

10 окт. 2015 г. · Bounded derivative implies uniform continuity on an open interval · Yes bounded derivatives necessarily do mean continuous on closed intervals.

28 февр. 2014 г. · It is well-known that if a differentiable function f:I→R (I an interval) has bounded derivative, then it is uniformly continuous.

22 нояб. 2023 г. · Bounded derivative is stronger than uniform continuity. A bounded derivative limits the slope, whereas, as you have discovered, a function may ...

16 окт. 2017 г. · A bounded derivative is a sufficient condition for uniform continuity, but not necessary. I know the counterexample f(x)= √ x on the interval [0,∞)

2 апр. 2015 г. · If the derivative is bounded, we have |f(x)|≤M for some M. By definition of a derivative, we have that f′(x)=limh→0|(f(x+h)−f(x)|h≤M.

12 нояб. 2014 г. · If we assume that the derivative of f is bounded, then f is uniformly continuous since it is in this case lipschitz continuous.

30 апр. 2013 г. · Suppose that f is a real-valued function on R whose derivative exists at each point and is bounded. Prove that f is uniformly continuous.

7 дек. 2017 г. · This is false. A function can be uniformly continuous even if there are places where its derivative is arbitrarily big; you just need those big ...

6 нояб. 2015 г. · Real-valued, differentiable function on R with the bounded derivative is uniformly continuous. Let f:R→R be a differentiable function such that there is M>0 ...