querent

Spam yourself with math PhD qualifying-exam questions.

UC Irvine Fall 2025, 4 Real analysis, Riemann integrability, Dirichlet-type functions

Define $𝑓 : [- 1, 1] \to ℝ$ by

𝑓 (𝑥) = {\begin{cases} sin 𝑥 & if 𝑥 \in ℚ \cap [- 1, 1] \\ 0 & if 𝑥 \notin ℚ \cap [- 1, 1] . \end{cases}

Determine whether $𝑓$ is Riemann integrable on $[- 1, 1]$ and justify the answer.

UC Irvine Fall 2025, 4 Real analysis, bounded variation, total variation, sup norm duality

Let $𝑓 \in 𝐶_{𝑐} (ℝ)$ . Show that there is a sequence $𝑔_{𝑛} \in BV (ℝ)$ with

TV (𝑔_{𝑛}) = 1

such that

lim_{𝑛 \to \infty} \int_{ℝ} 𝑓 (𝑥) 𝑔_{𝑛^{'}} (𝑥) d 𝑥 = sup_{𝑥 \in ℝ} | 𝑓 (𝑥) | = ∥ 𝑓 \underset{\infty}{∥} .

Stanford Fall 2025, 5 Real analysis, smooth functions, partitions of unity

(a) Prove that there is a $𝐶^{\infty}$ function $𝑓 : ℝ \to ℝ$ which vanishes on $(- \infty, 0]$ and is positive on $(0, \infty)$ .

(b) Let $𝑈 \subset ℝ^{𝑛}$ be an open set and $𝑥_{0} \in 𝑈$ . Prove that there is a $𝐶^{\infty}$ function $𝑔 : ℝ^{𝑛} \to ℝ$ such that $𝑔 (𝑥_{0}) = 1$ and that $supp (𝑔) \subset 𝑈$ .

(c) Let $⋃_{𝑖 = 1}^{𝑘} 𝑈_{𝑖}$ be an open cover of $⊤^{𝑛} = {(ℝ / (2 𝜋 ℤ))}^{𝑛}$ . Prove that for every $𝑖 = 1, \dots, 𝑘$ , there is a $𝐶^{\infty}$ function $𝜑_{𝑖} : ⊤^{𝑛} \to ℝ$ such that $𝜑_{𝑖} \geq 0$ and $supp (𝜑_{𝑖}) \subset 𝑈_{𝑖}$ so that $\sum_{𝑖 = 1}^{𝑘} 𝜑_{𝑖} = 1$ .

Stanford Fall 2025, 5 Real analysis, Sobolev spaces, Fourier transform, Hölder continuity

Fix $𝑠 \in (1 / 2, 1]$ . For a Schwartz function $𝑓 : ℝ \to ℂ$ , define

{‖ 𝑓 ‖}_{{\dot{𝐻}}^{𝑠} (ℝ)} = {(\int_{ℝ} {| 𝜉 |}^{2 𝑠} {| \hat{𝑓} |}^{2} (𝜉) d 𝜉)}^{1 / 2} .

(a) Prove that there exists $𝐶_{1} > 0$ such that whenever $𝑓 \in 𝒮︀ (ℝ)$ and $supp (\hat{𝑓}) \subset [- 𝐴, 𝐴]$ , then

{‖ 𝑓^{'} ‖}_{𝐿^{\infty} (ℝ)} \leq 𝐶_{1} 𝐴^{3 / 2 - 𝑠} {‖ 𝑓 ‖}_{{\dot{𝐻}}^{𝑠} (ℝ)} .

(b) Prove that there exists $𝐶_{2} > 0$ such that whenever $𝑓 \in 𝒮︀ (ℝ)$ and $supp (\hat{𝑓}) \subset ℝ ∖ [- 𝐴, 𝐴]$ , then

{‖ 𝑓 ‖}_{𝐿^{\infty} (ℝ)} \leq 𝐶_{2} 𝐴^{- 𝑠 + 1 / 2} {‖ 𝑓 ‖}_{{\dot{𝐻}}^{𝑠} (ℝ)} .

sup_{𝑥 \neq 𝑦} \frac{| 𝑓 (𝑥) - 𝑓 (𝑦) |}{{| 𝑥 - 𝑦 |}^{𝑠 - 1 / 2}} \leq 𝐶_{3} {‖ 𝑓 ‖}_{{\dot{𝐻}}^{𝑠} (ℝ)} .

All constants are allowed to depend on $𝑠$ .

UC Irvine Fall 2025, 5 Real analysis, Lipschitz continuity, gradient bounds, multivariable calculus

Let $𝑓 : ℝ^{2} \to ℝ$ be differentiable and suppose there exists a constant $𝑀 > 0$ such that

∥ \nabla 𝑓 (𝑥, 𝑦) ∥ \leq 𝑀

for all $(𝑥, 𝑦) \in ℝ^{2}$ . Prove that $𝑓$ is Lipschitz continuous, namely for all $(𝑥, 𝑦), (𝑎, 𝑏) \in ℝ^{2}$ ,

| 𝑓 (𝑥, 𝑦) - 𝑓 (𝑎, 𝑏) | \leq 𝑀 \sqrt{{(𝑥 - 𝑎)}^{2} + {(𝑦 - 𝑏)}^{2}} .

UC Irvine Fall 2025, 5 Real analysis, outer measure, Carathéodory criterion, measurability

Let $𝐸 \subset ℝ$ have finite outer measure. Prove that $𝐸$ is Lebesgue measurable if and only if for each open bounded interval $(𝑎, 𝑏)$ ,

𝑏 - 𝑎 = 𝑚^{*} ((𝑎, 𝑏) \cap 𝐸) + 𝑚^{*} ((𝑎, 𝑏) \ 𝐸) .

UC Irvine Fall 2025, 6 Real analysis, lower semicontinuity, epigraphs, minimum attainment

A function $𝑓 : ℝ \to ℝ$ is lower semi-continuous at $𝑎 \in ℝ$ if for all $𝜀 > 0$ , there exists $𝛿 > 0$ such that, for all $𝑥 \in ℝ$ , if $| 𝑎 - 𝑥 | < 𝛿$ , then

𝑓 (𝑥) \geq 𝑓 (𝑎) - 𝜀 .

The function $𝑓$ is lower semi-continuous if it is lower semi-continuous at every $𝑎 \in ℝ$ .

(a) Prove that $𝑓$ is lower semi-continuous if and only if

{(𝑥, 𝑦) \in ℝ^{2} : 𝑦 \geq 𝑓 (𝑥)}

is closed.

(b) If $𝑓_{𝑛} : ℝ \to ℝ$ are lower semi-continuous functions, show that

𝑓 = sup_{𝑛} 𝑓_{𝑛}

is also lower semi-continuous.

(c) If $𝑓$ is lower semi-continuous and bounded from below, show that $𝑓$ attains a minimum value on any interval $[𝑎, 𝑏]$ .

UC Irvine Fall 2025, 6 Real analysis, pointwise convergence, L^p convergence, uniform integrability

Assume that

\int_{0}^{1} | 𝑓_{𝑛} (𝑥) |^{3} d 𝑥 \leq 1, 𝑛 \in ℕ,

where $𝑓_{𝑛} : [0, 1] \to ℝ$ are functions such that

𝑓_{𝑛} (𝑥) \to 0

pointwise almost everywhere as $𝑛 \to \infty$ . Show that

∥ 𝑓_{𝑛} \underset{2}{∥} \to 0

as $𝑛 \to \infty$ .

UC Irvine Fall 2025, 7 Real analysis, Arzelà–Ascoli, uniform convergence, integral operators

Let ${𝑓_{𝑛}}$ be a sequence of continuous functions $[0, 1] \to ℝ$ such that for some $𝑀 > 0$ ,

\int_{0}^{1} | 𝑓_{𝑛} (𝑡) |^{2} d 𝑡 \leq 𝑀

for all $𝑛 \in ℕ$ . Define $𝑔_{𝑛} : [0, 1] \to ℝ$ by

𝑔_{𝑛} (𝑥) = \int_{0}^{1} \sqrt{𝑥 + 𝑡} 𝑓_{𝑛} (𝑡) d 𝑡 .

Show that there exists a subsequence of ${𝑔_{𝑛}}$ that converges uniformly on $[0, 1]$ .

UC Irvine Fall 2025, 8 Real analysis, Chebyshev's sum inequality, rearrangement inequalities

Let $𝑓 : [𝑎, 𝑏] \to ℝ$ be a continuous function. Prove that, for any non-decreasing continuous function $𝐺 : ℝ \to ℝ$ , we have

(\int_{𝑎}^{𝑏} 𝑓 (𝑥) 𝐺 (𝑓 (𝑥)) d 𝑥) (\int_{𝑎}^{𝑏} d 𝑥) \geq (\int_{𝑎}^{𝑏} 𝐺 (𝑓 (𝑥)) d 𝑥) (\int_{𝑎}^{𝑏} 𝑓 (𝑥) d 𝑥) .

Moreover, if $𝐺$ is strictly increasing, prove that equality holds if and only if $𝑓$ is a constant function.

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