Intereting Posts

Why is $L_A$ not $\mathbb K$ linear (I can prove that it is)
Show that $\pi =4-\sum_{n=1}^{\infty }\frac{(n)!(n-1)!}{(2n+1)!}2^{n+1}$
Prove that $L = \frac{(pX')' – qX}{r}$ is a formally self-adjoint operator for continuous $p, q,r$ functions.
If 2 open balls define the same space, is it true that x=y and r=s?
Demonstrate that every martingale is a local martingale.
Showing every knot has a regular projection using differential topology
Why we wonder to know all derivations of an algebra?
Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.
Reason for Continuous Spectrum of Laplacian
Finding a limit with a cube
Differential Equations with Deviating Argument
number of triangles determined by a rectangular grid
conversion of laplacian from cartesian to spherical coordinates
How is PL knot theory related to smooth knot theory?
If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$

Let $f$ be a continues function in $[a,b]$

$\forall x \in [a,b] \ \ \ f(x)\geq 0$

$ \int_{a}^{b}f(x)dx \ = 0$

- Find the limit: $\lim _{n\rightarrow +\infty }\sqrt {\frac {\left( 2n\right) !} {n!\,n^{n}}}$
- $\int_0^1 {\frac{{\ln (1 - x)}}{x}}$ without power series
- Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$
- Integral big question
- Why is the differentiation of $e^x$ is $e^x$?
- How to evaluate this limit? Riemann Integral

Proof that $ \forall x \in [a,b] \ \ f(x) = 0 $

So how do I do that ?

What I know is:

because $f$ is continues function in $[a,b]$ I know it bounded and because $\forall x \in [a,b] \ \ \ f(x)\geq 0$ I know that $ \exists M \in \mathbb{N} \ \forall x \in [a,b] \ \ \ M \geq f(x)\geq 0$.

I also know that because $f$ is continues function in $[a,b]$ that $f \in R[a,b]$ meaning that there exists $F$ so that $F(a) = F(b)$, but how can I show that $ \forall x \in [a,b] \ \ F(b) = F(a) = F(x)$ ?

So how do I continue from here ?

- Non-circular proof of $\lim_{\theta \to 0}\frac{\sin\theta}{\theta} = 1$
- Dirichlet's test for convergence of improper integrals
- find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges
- Proving a formula for $\int_0^\infty \frac{\log(1+x^{4n})}{1+x^2}dx $ if $n=1,2,3,\cdots$
- Integrating exponential of exponential function
- Limit theorem problem
- Is this a function and does F have an inverse?
- Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$
- Riemann-integrable (improperly) but not Lebesgue-integrable
- Integration of exponential and square root function

Suppose not. There exists a $x$ such that $f(x) > 0$. Let $r = f(x)$. Then by continuity, there exists a $\delta$ such that for all $y$ with $|x – y| < \delta$, $|f(x) – f(y)| = |r – f(y)| < \frac{r}{2}$. Hence for all $y \in (x – \delta, x + \delta)$, $f(y) > \frac{r}{2}$. Hence $\int_a^b f \ dx \geq \int_{x – \delta}^{x + \delta} f \ dx \geq \frac{r}{2} \cdot 2 \delta > 0$. This contradicts $\int_a^b f \ dx = 0$.

- For groups $A,B,C$, if $A\times B$ and $A\times C$ are isomorphic do we have $B$ isomorphic to $C$?
- How to properly set up partial fractions for repeated denominator factors
- How to find sum of power series $\sum_{n=0}^\infty\frac{1}{n!(n+2)}$ by differentiation and integration?
- Why is $A_5$ a simple group?
- Difference between $\mathbb{Q}/(X-1) \otimes_\mathbb{Q} \mathbb{Q}/(X+1)$ and $\mathbb{Q}/(X-1)\otimes_{\mathbb{Q}}\mathbb{Q}/(X+1)$?
- In naive set theory ∅ = {∅} = {{∅}}?
- Help with calculating infinite sum $\sum_{n=0}^{\infty}\frac1{1+n^2}$
- Prove projection is self adjoint if and only if kernel and image are orthogonal complements
- Is it possible to gain intuition into these trig compound angle formulas – and in general, final year high school math?
- mutual information of coupled variables
- Computing the characteristic function of a normal random vector
- Is a faithful representation of the orthogonal group on a vector space equivalent to a choice of inner product?
- induced representation, dihedral group
- For a set of symmetric matrices $A_i$ of order p, show that if the sum of their ranks is p, $A_iA_j=0$
- Verifying that $\lim_{\alpha\to \infty} \frac{1}{\pi} \frac{\sin^2\alpha x}{\alpha x^2}= \delta(x)$