Real Analysis II · Differentiation
Dense zero set for a continuous derivative
Problem
Let I be an interval and f:I→ℝ be differentiable. If f' is continuous on I and vanishes on a dense subset of I, prove f is constant on I.
Hint
Use density to choose points of the zero set converging to an arbitrary point of the interval.
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
Create a free account to check your answer and see the solution. Create a free account.
More Real Analysis II practice problems
- Cauchy criterion for sumsIntegration
- Two-jump integratorIntegration
- Dense discontinuities with integrabilityIntegration
- Derivative interchange theoremUniform Convergence
- Uniform convergence of distribution functionsUniform Convergence
- Abel regularization of an absolutely convergent Fourier seriesPower Series & Capstones
- Neumann series integral equationPower Series & Capstones
- A dense-set function differentiable only at the originDifferentiation
- Classic tangent cancellationDifferentiation