$x \mapsto x^2$ on $[0, 1]$

Terms

Term	This example	Why
riemann integral	$\int_0^1 x^2 \mathrm{d}x = \frac{1}{3}$	The Riemann integral of $x^2$ on $[0, 1]$ is $\frac{1}{3}$, because $\frac{x^3}{3}$ is an antiderivative.
lebesgue integral	$\frac{1}{3}$	The Lebesgue integral of $x^2$ on $[0, 1]$ equals its Riemann integral, as for every Riemann integrable function.
locally Lipschitz continuous	✓	$x \mapsto x^2$ on $[0, 1]$ is locally Lipschitz continuous, because the function is Lipschitz continuous on all of $[0, 1]$ with constant $2$.
Lipschitz continuous	✓	$x \mapsto x^2$ on $[0, 1]$ is Lipschitz continuous because that derivative is at most $2$ on $[0, 1]$.
continuously differentiable	✓	$x \mapsto x^2$ on $[0, 1]$ is continuously differentiable because its derivative $2x$ is continuous.
absolutely continuous	✓	$x \mapsto x^2$ on $[0, 1]$ is absolutely continuous, because it is Lipschitz continuous.
continuous	✓	$x \mapsto x^2$ on $[0, 1]$ is continuous.
uniformly continuous	✓	$x \mapsto x^2$ on $[0, 1]$ is uniformly continuous because every continuous function on a compact set is uniformly continuous.