Intereting Posts

A criterion for independence based on Characteristic function
Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$
Groups of order $56$
How do you show that $f(x) = e^{-x^2}$ is in the Schwartz space $\mathcal{S}(\Bbb{R})$?
Groups that are not Lie Groups
Summation using residues
How to make the encoding of symbols needs only 1.58496 bits/symbol as carried out in theory?
Double Factorial: Number of possibilities to partition a set of $2n$ items into $n$ pairs
square cake with raisins
Show that $\int_0^1\ln(-\ln{x})\cdot{\mathrm dx\over 1+x^2}=-\sum\limits_{n=0}^\infty{1\over 2n+1}\cdot{2\pi\over e^{\pi(2n+1)}+1}$ and evaluate it
Unique perpendicular line
What could the ratio of two sides of a triangle possibly have to do with exponential functions?
What Does This Notation Mean (“derivative” of a 1-form)?
What are the primitive notions of real analysis?
Block Diagonal Matrix Diagonalizable

This question is an exact duplicate of:

- Correlation between the weak solutions of a differential equation and implied differential equations

- Prove that $ \int \limits_a^b f(x) dx$ = $ \int \limits_a^b f(a+b-x) dx$
- Is $ \frac{\mathrm{d}{x}}{\mathrm{d}{y}} = \frac{1}{\left( \frac{\mathrm{d}{y}}{\mathrm{d}{x}} \right)} $?
- Proving $f'(1)$ exist for $f$ satisfying $f(xy)=xf(y)+yf(x)$
- An inflection point where the second derivative doesn't exist?
- Rewriting the time-independent Schrödinger equation for a simple harmonic oscillating potential in terms of new variables
- Lipschitz continuity implies differentiability almost everywhere.
- Proving $f(x) = x^2 \sin(1/x)$, $f(0)=0$ is differentiable at $0$, with derivative $f'(0)= 0$ at zero
- We can define the derivative of a function whose domain is a subset of rational numbers?
- Find the equation of the tangent line to the curve at the given point. $y = 1+2x-x^3$ at $(1,2)$
- Differentiability of an homogeneous and continuous function $f$ ($f(\alpha x)=\alpha^\beta f(x)$)

One direction is trivial: a solution to any ODE is clearly a continuous solution to the corresponding IDE (implied differential equation).

The opposite direction seems to be false. Consider $I(f)=2\lfloor x\rfloor -1$ on the interval $[-\frac12, \frac12]$. This is solved by $f(x)=|x|$, which is continuous. But $f’=2\lfloor x\rfloor -1$ has no solutions: since RHS has a nonessential discontinuity at 0, it cannot be the derivative of any function.

The solution below doesn’t work because Darboux’s theorem assumes $f$ is differentiable. I’ve kept it for archival purposes.

When we restrict our attention to functions defined on a closed interval, we get a simple solution thanks to Darboux’s theorem. This theorem implies, in particular, that $f’$ has only essential discontinuities (this fact is mentioned on the Wiki page currently). But the fact that $f$ is a solution to an IDE on the interval implies that it has only nonessential discontinuities.

So in fact, $f’$ is continuous on the whole interval, which means that $f$ is differentiable, which means that $I(f)=D(f)$ (or, more pedantically, $I(f)(a)=\{D(f)(a)\}$ for all $a$ in the interval). This is certainly strong enough to prove the desired result ðŸ˜›

I will spend some time thinking about more exotic scenarios. ~~In particular, I have reason to believe that OP would be ~~ (we talked in chat; this isn’t true). I am not quite sure how derivatives at endpoints work, so if there anyone can shore up that bit in a line or two, please mention it in the comments ðŸ™‚*particularly* interested in the case of half-open intervals

- Roadmap to study Atiyahâ€“Singer index theorem
- On a system of equations with $x^{k} + y^{k} + z^{k}=3$ revisited
- how to prove $a+b-ab \le 1$ if $a,b \in $?
- Show that $\sqrt{n+1}-\sqrt{n}\to0$
- A non-Vandermonde matrix with Vandermonde-like determinant?
- find the measure of $AMC$
- $X$ Poisson distribution, $Y$ geometric distribution – how to find $P(Y>X)$?
- Connectedness of sets in the plane with rational coordinates and at least one irrational
- Equation of angle bisector, given the equations of two lines in 2D
- Showing that $1/x$ is NOT Lebesgue Integrable on $(0,1]$
- Is there a rule of integration that corresponds to the quotient rule?
- Comaximal ideals
- How was the Fourier Transform created?
- Axiom of Choice: Can someone explain the fallacy in this reasoning?
- Generalizing Ramanujan's cube roots of cubic roots identities