Articles of real analysis

Is $x^{3/2}\sin(\frac{1}{x})$ of bounded variation?

I’m trying to show that $f(x)$ is of bounded variation where $f(x)=x^{3/2}\sin(\frac{1}{x})$ on $[0,1]$. I think that it is but I can’t show it explicitly. Any help will be appreciated.

Prove $(n + 1)I(n) = (n + 2)I(n + 2)$ by integration by parts.

$I_n$ is given by $\int ^ {\pi/2}_{0} \sin^n(x) dx$ My attempt: I got to the fact that the statement is true iff $$\dfrac {n+1}{n+2} = \int ^ {\pi/2}_{0} \dfrac {\sin^2(x)}{\sin^n(x)}$$ I do not know how to prove the above statement using integration by parts.

$\displaystyle 3^x+4^x=5^x$

This question already has an answer here: Proving that $ 2 $ is the only real solution of $ 3^x+4^x=5^x $ 2 answers

continuous partial derivative implies total differentiable (check)

Let $f: \mathbb{R^n}\rightarrow \mathbb{R}$ to have continuous partial derivatives, it suffice to show that $f$ is total differentiable at $(0,..,0)$ with $f_{x_i}(0,..,0) = 0$. Since each partial derivative is continuous, we can rewrite $f$ as $$f(0,..,0) + \int_0^{x_1} f_{x_1}(t_1,0,..,0) dt_1 + \int_0^{x_2} f_{x_2}(x_1,t_2,0,..,0) dt_2 + … +\int_0^{x_n} f_{x_n}(x_1,x_2,.., t_n) dt_n,$$ and to show $$\lim_{x\rightarrow 0; […]

Does a function have to be bounded to be uniformly continuous?

My book defines uniform continuity as a form of continuity that works for any points $a$ and $x$ in an interval $I$ such that $$|x-a| < \delta$$ implies that $$f(x) – f(a) < \epsilon$$ It then goes on to assert that “If $f$ is continuous over a closed and bounded interval $[a,b]$, it is uniformly […]

Laplacian as a Fredholm operator

Let $\Omega$ be a bounded smooth domain in $\mathbb R^n$. The Laplacian $\Delta$ acts on functions on $\Omega$. From elliptic regularity (I haven’t worked out all the details), we have that $$ \Delta : C^{2,\alpha} (\Omega) \to C^{0, \alpha}(\Omega),\ \ \alpha >0$$ and $$\Delta : W^{k,p}(\Omega) \to W^{k-2, p} (\Omega),\ \ k\ge 2, p \in […]

Mean Value property for harmonic functions on regions other than balls/spheres

Let $u$ be a harmonic function on $\mathbb R^n$. We know that if $B$ is a ball centered at $x$, then $$u(x) = \frac{1}{|B|} \int_B u(y) dy = \frac{1}{|\partial B|} \int_{\partial B} u(z) dS(z).$$ I am wondering if there is an analagous result with the ball $B$ replaced by a different set, perhaps an $n$-cube […]

If $f''(x)\gt 0$ and $f'(x)\gt 0$, then $\lim_{x\to +\infty}f(x)=+\infty$

I’m trying to solve this question: Suppose $f”(x)\gt 0$ in $(a,+\infty)$ and there is $x_0\gt a$ such that $f'(x_0)\gt 0$. Prove that $\lim_{x\to +\infty}f(x)=+\infty$. My attempt: I know that there is $\alpha \in (a,x_0)$ such that $f”(\alpha)=\frac{f'(x_0)-f'(a)}{x_0 – a}\gt0$, then $f'(x_0)\gt f'(a).$ I don’t know what to do with this information, I can’t follow from […]

Test for the convergence of the sequence $S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$

$$S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$$ Show the convergence of $S_n$ (the method of difference more preferably) I just began treating sequences in school, and our teacher taught that monotone increasing sequence, bounded above and monotone decreasing sequences, bounded below converge. and so using that theorem here.. I found the $$(n+1)_{th} […]

Limit of the derivative and LUB

Let $(k,+,.,0,1,<)$ be an ordered field. In the folowing definitions, all numbers and notions are derived from the ordered field structure of $k$, and $a < c$ are generic elements of $k$. Define the limit of a function $f: ]a;b[ \cup ]b;c[ \longrightarrow k$ at $b$ as the element $l$ of $k$ if there is […]