Intereting Posts

Every manifold is locally compact?
What is exactly the difference between $\forall x \neg P(x)$ and $\neg \forall xP(x)$?
there exist some real $a >0$ such that $\tan{a} = a$
Interpretations of the first cohomology group
Which sequences converge in a cofinite topology and what is their limit?
show that $f(z)+f(z^2)+\cdots + f(z^n)+\cdots$ converges locally uniformly to an analytic function in the unit disk.
Prove 24 divides $u^3-u$ for all odd natural numbers $u$
Is an automorphism of the field of real numbers the identity map?
Does $\sin(t)$ have the same frequency as $\sin(\sin(t))$?
$\lim_n \frac{1}{n} E(\max_{1\le j\le n} |X_j|) = 0$
Prove that $\sqrt{x}$ is continuous on its domain $[0, \infty).$
Proof of $\arctan{2} = \pi/2 -\arctan{1/2}$
Every proper subspace of a normed vector space has empty interior
Consistency/range conditions for (integral) transform mapping into higher-dimensional space
Why did the ancients hate the Parallel Postulate?

Suppose $f$ is a continuous function over $[0,1]$ such that $f(0) = f(1).$ Show that for any positive integer $n$ there is $x \in [0,1-\frac{1}{n}]$ for which $f(x) = f(x+\frac{1}{n})$.

We seem to be saying that $f(x+\frac{1}{n})$ is periodic with respect to any positive integer $n$. Also this seems to make sense since we start and end at the same spot there have to be values of $x$ with the same $f(x)$ as other $x$’s. But I am struggling to see how to prove this specific statement.

- First year calculus student: why isn't the derivative the slope of a secant line with an infinitesimally small distance separating the points?
- Integral $\int_0^1 \ln(x)^n \operatorname{Ei}(x) \, dx$
- Evaluate or simplify $\int\frac{1}{\ln x}\,dx$
- Difference between a Gradient and Tangent
- Fundamental Theorem of Calculus: Why Doesn't the Integral Depend on Lower Bound?
- How to integrate this improper integral.

- How to find the function $f$ given $f(f(x)) = 2x$?
- What is infinity divided by infinity?
- Optimization, volume of a box
- Evaluation of $\int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx$
- If we define $\sin x$ as series, how can we obtain the geometric meaning of $\sin x$?
- Show that equation has no solution in $(0,2\pi)$
- Integral $\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx$
- Integral $\int_0^{\large\frac{\pi}{4}}\left(\frac{1}{\log(\tan(x))}+\frac{1}{1-\tan(x)}\right)dx$
- Integrating $\int_0^{\pi/2} \cos^a(x) \cos(bx) \ dx$
- To show that $e^x > 1+x$ for any $x\ne 0$

Consider $g(x) = f(x) – f(x+\frac{1}{n})$. We want to prove that $g(x) = 0$ for some $x\in [0,1-\frac{1}{n}]$. Consider the values $g(0), g(1/n), g(2/n), \ldots, g(1-1/n)$. At least one of these values must be negative and at least one must be positive (if one is zero, we’re done). Otherwise, they are all positive or all negative. But the first case would mean $f(0) > f(1/n) > f(2/n) > \cdots > f(1-1/n) > f(1)$ which is false, and the second case would mean $f(0) < f(1)$. Thus, one of the values is positive and one is negative, and therefore by continuity $g(x) = 0$ for some $x$ in the given interval.

- Showing that every finitely presented group has a $4$-manifold with it as its fundamental group
- Cardinality of cartesian product of an infinite set with N
- Can Spectra be described as abelian group objects in the category of Spaces? (in some appropriate $\infty$-sense)
- How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?
- $L$-function of an elliptic curve and isomorphism class
- Proving $\sqrt{ab} = \sqrt a\sqrt b$
- Why does the Dedekind Cut work well enough to define the Reals?
- Prove by induction Fibonacci equality
- Matrix Norm set #2
- Cyclic Group Generators of Order $n$
- Infinite Set is Disjoint Union of Two Infinite Sets
- “Mathematical Induction”
- Question of remainder on dividing by 7
- Other challenging logarithmic integral $\int_0^1 \frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx$
- Counting Functions or Asymptotic Densities for Subsets of k-almost Primes