Articles of real analysis

Integral of an increasing function is convex?

Let $f$ be a real valued differentiable function defined for all $x \geq a$. Consider a function F defined by $F(x) = \int_a^x f(t) dt$. If f is increasing on any interval, then on that interval F is convex. I am not sure I intuitively understand this. What is the function is increasing at an […]

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This question already has an answer here: How does the existence of a limit imply that a function is uniformly continuous 5 answers

Real analysis supremum proof

Let A be a non-empty bounded sub-set of $\mathbb{R}$. Let $B\subset\mathbb{R}$, given by:$B=\{\frac{a_1+2a_2}{2}|a_1,a_2\in A\}$. Express $\sup B$ in terms of $\sup A$. My attempt: Suppose $a_1,a_2\in A$ and $b\in B$. Then $a_1 \leq \sup (A)$ and $a_2\leq \sup (A)$ So $a_1 + 2a_2\leq 3sup (A)$. This gives $\frac{a_1 + 2a_2}{2}\leq \frac{3\sup (A)}{2}$. This means that […]

Finding the heat flow across the curved surface of a cylinder

I have the following problem: The temperature at a point in a cylinder of radius $a$ and height $h$, and made of material with conductivity $k$, is inversely proportional to the distance from the centre of the cylinder. Find the heat flow across the curved surface of the cylinder. The solution says that $T = […]

Energy for the 1D Heat Equation

So consider the heat equation on a rod of length $L$, $u_t (x,t) = c^2 u_{xx} (x,t)$, $\forall (x,t) \in [0,L]$ x $\mathbb{R}^+ $, and the energy at time $t$ defined as, $$E(t)=\frac{1}{2}\int_{0}^{L} u(x,t)^2 dx.$$ How would I show that $E(t) \geq 0$ for every $t \in \mathbb{R}^+$, and that $$ E'(t) = -c^2 \int_{0}^{L} […]

Question on using sandwich rule with trig and abs function to show that a limit exists.

Past paper Question: For the following function, determine whether $\lim_{x\to\infty}f(x)$ exists, and compute the limit if it exists. Justify your answers. $$f(x)= \dfrac{\sin(x)+1}{\left| x \right|}$$ Attempt: Consider the fact that $-1 \le \sin(x) \le 1$ (for all $x$), which implies $0 \le \sin(x) +1\le 2$. Dividing by $\left| x \right|,$ $$\color{green}{ \frac{0}{\left| x \right|}} \le […]

An open interval as a union of closed intervals

For $a<b, a,b\in\Bbb R$ $$(a,b)=\bigcup_{0<\delta<(b-a)/2} I_{\delta} \quad I_{\delta}:=[a+\delta,b-\delta] $$ Clearly the RHS is an (uncountable) infinite sum of closed intervals. I have no idea how to show it is open at two ends. (My hope is that if this is true, then it is trivial that: $$f(x)\in\mathscr C^r(a,b)\Longleftrightarrow f(x)\in\mathscr C^r[a+\delta,b-\delta]\quad \forall \delta\in(0,\frac{b-a}2) $$ which will […]

Prove that if $\lim _{x\to \infty } f(x)$,then $\lim_{x\to \infty} f(x)=0$

Let $f:\Bbb R\to \Bbb R$ be a continuous function such that $\int _0^\infty f(x)\text{dx}$ exists. Prove that if $\lim _{x\to \infty } f(x)$,then $\lim_{x\to \infty} f(x)=0$ If $f$ is non-negative then $\lim _{x\to \infty } f(x)$ must exist and $\lim_{x\to \infty} f(x)=0$ My try To prove that $\lim_{x\to \infty} f(x)=0$ we should show that $\exists […]

How to prove that no constant can bound the function f(x) = x

I know this is a trivial question, but how would one mathematically demonstrate this using a proof?

Lipschitz in $\mathbb R^1$ implies Lipschitz along any line in $\mathbb R^k$ (for convex functions)

I’m trying to solve a homework problem (baby rudin course) and realized that Lipschitz continuity really makes the problem easier for me. However, since it is not defined in Rudin I have to prove the properties if I want to use them. I have just argued that convex functions are (locally) lipschitz in $\mathbb R^1$. […]