Intereting Posts

do all uncountable sets have same cardinality as real numbers?
How to calculate the following conditional expectation? Is my calculation process right?
Prove $\sum_{k=1}^{\infty} \frac{\sin(kx)}{k} $ converges
Proof of the polynomial division algorithm
What are the subgroups of a semidirect product?
Why $\mathbf Z/(2, 1+\sqrt{-5})\simeq \mathbf Z/(2,x+1,x^2+5)$?
Improper Integral $\int\limits_0^1\frac{\ln(x)}{x^2-1}\,dx$
Catalan numbers – number of ways to stack coins
Properties of the countable complement topology on $\mathbb{R}$.
Summing divergent series based on physics
If D is an Integral Domain and has finite characteristic p, prove p is prime.
A bound on the derivative of a concave function via another concave function
Find n that : $1+5u_nu_{n+1}=k^2, k \in N$
Expanded concept of elementary function?
A proof that powers of two cannot be expressed as the sum of multiple consecutive positive integers that uses binary representations?

To prove $\tan(x)$ defined on $]-π/2;π/2[$ is injective I take the derivative of $\tan(x)$ to get $\sec(x)^2$.

This shows that $\tan(x)$ is monotonic (strictly) increasing which implies it is injective.

However how do I show it is surjective ? That every single real number corresponds to some number in the domain of $\tan(x)$ ?

- Can I bring the variable of integration inside the integral?
- How to prove that if $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+…+a_n}{n}=A$
- Why aren't all dense subsets of $\mathbb{R}$ uncountable?
- Pointwise supremum of a convex function collection
- Can we determine uniform continuity from graphs?
- Lipschitz continuity of atomless measures

Thanks in advance

- What are the zero divisors of $C$?
- Why do we want TWO open sets from the inverse function theorem?
- When can we exchange order of two limits?
- An exercise from Stein's Fourier analysis about wave equation
- Covariant and partial derivative commute?
- $C^\infty$ approximations of $f(r) = |r|^{m-1}r$
- Expressing $\Bbb N$ as an infinite union of disjoint infinite subsets.
- A continuous mapping is determined by its values on a dense set
- Can an infinite sum of irrational numbers be rational?
- Is $ \sum\limits_{n=1}^\infty \frac{|\sin n|^n}n$ convergent？

I can see that a rigorous analytical proof for the surjectivity of $f\colon(-\pi/2,\pi/2)\to\mathbb{R}$, where $f(x)=\tan(x)$, is a way far off. However, I wanted to give a picture of what @achillehui mentioned in a comment, as it is rather beautiful in my opinion:

$\color{white}{Put it in the center!!!!!!!}$

Given the picture above, we note that

$$

\tan(\theta)=\frac{x}{1}=x\qquad\text{and}\qquad\tan(-\theta)=-\frac{x}{1}=-x.

$$

Hence, we can “clearly” see that

$$

\lim_{\theta\to\pi/2^-}\tan(\theta)=\infty\qquad\text{and}\qquad\lim_{-\theta\to-\pi/2^+}\tan(-\theta)=-\infty.\qquad\approx\blacksquare

$$

- Why does $\int_0^\infty\frac{\ln (1+x)}{\ln^2(x)+\pi^2}\frac{dx}{x^2}$ give the Euler-Mascheroni constant?
- How to show every subgroup of a cyclic group is cyclic?
- Weak convergence of random variables
- Verifying $|F(r)| \geq \frac{1}{1-r}\log(\frac{1}{1-r}) $ and $|F(re^{i \theta})| \geq c_{q/r}\frac{1}{1-r}\log({\log(\frac{1}{1-r})})$
- Find that solution $ϕ$ which satisfies $ϕ(0) = 1, ϕ'(0) = 2$ for $y'' = 3x + 1$.
- Holomorph is isomorphic to normalizer of subgroup of symmetric group?
- Prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded.
- Prove by contradiction that a real number that is less than every positive real number cannot be posisitve
- Anyone has a good recommendation of a free pdf book on group theory?
- How to Compute Genus
- Chicken Problem from Terry Tao's blog (system of Diophantine equations)
- Weak topology and the topology of pointwise convergence
- Visibility and Kernel of Polygon
- Examples of a monad in a monoid (i.e. category with one object)?
- Prove: The product of any three consecutive integers is divisible by $6$.