Intereting Posts

Can you please check my Cesaro means proof
Interesting integral formula
Is $e^e$ irrational?
How to Decompose $\mathbb{N}$ like this?
Show $R \setminus S$ is a union of prime ideals
finding Expected Value for a system with N events all having exponential distribution
Number of classes of k-digit strings when digit order and identity doesn't matter
Exercise from Serre's “Trees” – prove that a given group is trivial
Evaluate $\int_0^1 \frac{x^k-1}{\ln x}dx $ using high school techniques
Complex power of complex number
Divisibility criteria for $7,11,13,17,19$
Good book for high school algebra
The number of ring homomorphisms from $\mathbb{Z}_m$ to $\mathbb{Z}_n$
What are some usual norms for matrices?
Stronger Nakayama's Lemma

The function $g$ is strictly positive. Let the function $f$ be defined as

$$f(x) = \int_0^x g(u) du$$

Is there a way to express $f^{-1}(x)$ in terms of $g$?

- Useful examples of pathological functions
- Sum of $\Gamma(n+a) / \Gamma(n+b)$
- Show $\lim\limits_{h\to 0} \frac{(a^h-1)}{h}$ exists without l'Hôpital or even referencing $e$ or natural log
- Evaluate $\sum_{k=1}^\infty \frac{k^2}{(k-1)!}$.
- How prove there is no continuous functions $f:\to \mathbb R$, such that $f(x)+f(x^2)=x$.
- A inverse Trigonometric multiple Integrals

- Interesting limit involving gamma function
- Global maximum and minimum of $f(x,y,z)=xyz$ with the constraint $x^2+2y^2+3z^2=6$ with Lagrange multipliers?
- Why isn't the directional derivative generally scaled down to the unit vector?
- “Box With No Top” Optimization
- Integration of exponential and square root function
- Proof of a Limit of a Function, given the Limits of the Multiplicative Inverses of the Function
- Comparison between integrals
- A function for which one-sided limits are zero and infinity
- Prove $kf(x)+f'(x)=0 $ when conditions of Rolle's theorem are satisfied .
- Multivariable calculus - Implicit function theorem

If $g$ takes on both negative and positive values, or is zero on some interval, then $f$ is not invertible, as mentioned in comments.

Assume $g$ is strictly positive (or strictly negative), hence $f^{-1}$ exists and is differentiable by inverse function theorem.

Then $f(f^{-1}(x))=x$, so by differentiating, we get that $f'(f^{-1}(x))(f^{-1})'(x) =1$, i.e, $g(f^{-1}(x))(f^{-1})'(x) = 1$.

Thus we see that $f^{-1}$ satisfies the differential equation $$y’ = \frac{1}{g(y)}$$

For some functions $g$ this can be solved exactly (for example, $g(y) = e^y$ or $g(y) = y^2+1$), while for others it cannot be solved exactly (for example $g(y) = e^{y^2}$). Hence, you can get this differential formula but no explicit solution in general.

- What is a support function: $\sup_{z \in K} \langle z, x \rangle$?
- If $f$ is midpoint convex, continuous, and two times differentiable, then for any $a, b \in \mathbb{R}$, there exists $c$ such that $f''(c) \geq 0$
- Horizontal tank with hemispherical ends depth to capacity calculation
- Is it always true that $\sum^{\infty}a_{i}1_{A_{i}}-\sum^{\infty}b_{i}1_{B_{i}}=\sum^{\infty}c_{i}1_{C_{i}}$?
- sum of irrational numbers – are there nontrivial examples?
- Boolean algebras without atoms
- Reduced schemes and global sections
- limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ and $a_1=1$
- Multiplying Taylor series and composition
- When does intersection of measure 0 implies interior-disjointness?
- A generalization of a divisibility relation for Fibonacci numbers
- Creating unusual probabilities with a single dice, using the minimal number of expected rolls
- eigen decomposition of an interesting matrix (general case)
- Constructing a number not in $\bigcup\limits_{k=1}^{\infty} (q_k-\frac{\epsilon}{2^k},q_k+\frac{\epsilon}{2^k})$
- On a remarkable system of fourth powers using $x^4+y^4+(x+y)^4=2z^4$