Real Analysis II · Differentiation
A dense-set function differentiable only at the origin
Problem
Define $g(x)=x^4$ when $x$ is rational and $g(x)=0$ when $x$ is irrational. Prove that $g$ is continuous only at $0$, is differentiable at $0$, and satisfies $g'(0)=0$.
Hint
For every $x$, $|g(x)|\le|x|^4$.
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
- Dense zero set for a continuous derivativeDifferentiation
- 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
- Classic tangent cancellationDifferentiation