Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 1 / 12E (Analysis I)

False. Every interval contains rationals and irrationals, so every lower Darboux sum is $0$ and every upper sum is $1$.

False. Thomae’s function is Riemann integrable and discontinuous at every rational, an infinite countable set.

False. Continuous functions can converge pointwise to the non-Riemann-integrable Dirichlet function; for example enumerate the rationals and use continuous narrow bumps whose first $n$ peaks cover the first $n$ rationals while widths shrink suitably pointwise off $\mathbb{Q}$.

False without continuity. The function equal to $1$ at one point and $0$ elsewhere is nonnegative, Riemann integrable with integral $0$, but is not identically zero.

False. The Cantor function has derivative $0$ outside the Cantor set, hence its derivative extends to the Riemann-integrable zero function, while $F(1)-F(0)=1$.

True. If $|f(x_0)|>C$, continuity gives an interval of positive length on which $|f|>C+\delta$. Its $L^p$ norm exceeds $(C+\delta ){\ell }^{1/p}$, which tends to $C+\delta >C$ as $p\to \infty$, a contradiction.