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Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

145439 questions
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Prove divergence and conclude that there is no universal "smallest" comparison series to test divergence

This is a practice problem for a midterm in a real analysis undergrad class, but I tagged it as homework anyway. Suppose that $a_k>0$ and that the series $\sum_{k=1}^\infty a_k$ diverges. Let $S_n$ = $\sum_{k=1}^n a_k$ and define $b_1 = a_1$ and…
David Meehan
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Let $f$ be a differentiable function from a closed interval to $\mathbb{R}$. Then, is $f'$ bounded?

Let $f$ be a differentiable function from a closed interval to $\mathbb{R}$. Then, $f'$ is bounded. Can somebody give a proof or a counterexample for this? Moreover, if $f$ is differentiable, then is $f'$ continuous?
yhk
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Question about discontinuous functions

While it is true that the product and addition/subtraction of continuous functions is continuous, is it true that the product and addtion/subtraction of discontinuous functions is also discontinuous? Edit Thanks to Chris Eagle I now understand the…
analysisj
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$f:\mathbb{R} \to \mathbb{R}$ st $f(\frac{1}{2^n})=0 \forall n \in \mathbb{N}$ , show $f'(0)=f''(0)=0$

Let$f:\mathbb{R} \to \mathbb{R}$ be twice differentiable function such that $f(\frac{1}{2^n})=0 \forall n \in \mathbb{N}$. Show that $f'(0)=f''(0)=0$ Applying mean value theorem on $(\frac{1}{2^n},\frac{1}{2^{n-1}})$ we see that $\exists c_n$ st…
Mathronaut
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Existence of antiderivative

Let $f$ be a real-valued function defined on the open unit interval. What assumptions you have to make about $f$ to be sure that it posseses an antiderivative? I'm interested in the weakest (most general) possible assumptions, so some nontrivial…
user150316
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Why is continuity of individual functions essential in Dini's theorem?

THEOREM: Suppose $\{f_n\}$ is a sequence of continuous functions from $[a,b]$ to $\Bbb R$ that converge pointwise to a continuous function $f$ over $[a,b]$. If $f_{n+1}\leq f_n$, then convergence is uniform. Then, why is the continuity of the the…
MathMan
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understanding the difference between uniform continuity and continuity

We have defined continuity as $\forall \epsilon, \exists \delta > 0 s.t. |x-c| < \delta, x \in D \implies |f(x) - f(c)| < \epsilon$ and uniform continuity as $\forall \epsilon, \exists \delta > 0 s.t. |x-y| < \delta, x,y \in D \implies |f(x) - f(y)|…
user144464
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$L^1$-convergence implies $L^p$-convergence

If $\{f_n\}$ is a sequence in $L^1$ and in $L^p$ where $p>1$, also $\{f_n\}$ converges to $f$ in $L^1$, does that imply $\{f_n\}$ converges to $f$ in $L^p$ as well? Thank you.
Salih Ucan
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Show $\exists x$ such that $Df(x) = 0$ for $f = 0$

Let $f:\mathbb{R}^n \to \mathbb{R}$ be continuously differentiable, $\Omega \subseteq \mathbb{R}^n$ an open and bounded set and $f = 0$ on $\partial \Omega$. Show that then there exists a $x \in \Omega$ such that $Df(x) = 0$. I dont't understand…
sj134
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Is there a base in which $1 + 2 + 3 + 4 + \dots = - \frac{1}{12}$ makes sense?

Consider the corresponding case for decimal. $9 + 1 = 10$ $99 + 1 = 100$ $999 + 1 = 1000$ ... We can convince ourselves that $ \overline{9}9 + 1 = 0$ or $-1 = \;...999$ In fact, this can be formalized in the case of floating point arithmetic or…
cactus314
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If $\sum a_n r^n$ converges, then $\sum a_n x^n$ converges uniformly in $[0,r]$

Hello this problem I have no idea what can I do. Let $\sum a_n x^n$ be a power series with finite convergence radius $r$. Prove that if $\sum a_n r^n$ converges, then $\sum a_n x^n$ converges uniformly in $[0,r]$.
August
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Definition of a twice differentiable function of two variables

Let's concider two definitions of twice differentiability: Definition 1. $f(x,y)$ is twice differentiable at $(x_0,y_0)$ iff a)$f^\prime_x, f^\prime_y$ are differentiable functions of two variables at $(x_0,y_0)$, b) $f$ is differentiable in a…
Alexander Kuleshov
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Is this result true? Double integrals

Let $A$ be a rectangle in $\mathbb{R}^2$. Suppose $f: A \to \mathbb{R}$ is bounded. Suppose $f(x) = 0 $ except on $F$, where $F$ is closed and has measure zero. Does it follow that $f$ is integrable on $A$ and $\int_A f = 0 $ ? Update with my…
user145801
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Prove complete metric space

Denote by $C^{\infty}(\mathbb{R})$ the space of infinitely differentiable functions. Prove that $C^{\infty}(\mathbb{R})$ is a complete metric space with respect to the metric…
Idonknow
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verify equicontinuous functions

Suppose $f_n(x)=\sin(nx)$ on $[0.2\pi]$. Is the sequence of functions $\{f_n\}_{n \geq 1}$ equicontinuous on $[0.2\pi]$? I try to compute derivative, which is $$f^{\prime}_n(x)=n\cos(nx)$$ which gives us $\sup_{x\in [0,2\pi]}|f^{\prime}_n(x)|=|n…
Idonknow
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