Related papers: The Implicit Function Theorem for maps that are on…

This article presents an elementary proof of the Implicit Function Theorem for differentiable maps F(x,y), defined on a finite-dimensional Euclidean space, with $\frac{\partial F}{\partial y}(x,y)$ only continuous at the base point. In the…

In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of…

This article presents simple and easy proofs of the Implicit Function Theorem and the Inverse Function Theorem, in this order, both of them on a finite-dimensional Euclidean space, that employ only the Intermediate Value Theorem and the…

The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…

This article is centered around generalizing a previous implicit function theorem of the author to be applicable for maps f:E sqcap F to F which can be lifted to Keller C^k_pi maps f_i:E sqcap F_i to F_i with F_i Banach and F=projlim F_i .…

We know that a continuous function on a closed interval satisfies the Intermediate Value Property. Likewise, the derivative function of a differentiable function on a closed interval satisfies the IVP property which is known as the Darboux…

We prove an implicit function theorem for Keller C^k_c-maps from arbitrary real or complex topological vector spaces to Frechet spaces, imposing only a certain metric estimate on the partial differentials. As a tool, we show the…

We introduce a notion of a noncommutative function defined on a domain of $d$-tuples of bounded operators on an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these…

The Implicit and Inverse Function Theorems are special cases of a general Implicit/Inverse Function Theorem which can be easily derived from either theorem. The theorems can thus be easily deduced from each other via the generalized…

We discuss some surprising phenomena from basic calculus related to oscillating functions and to the theorem on the differentiability of inverse functions. Among other things, we see that a continuously differentiable function with a strict…

In this article, we present explicit estimates of the size of the domain on which the Implicit Function Theorem and the Inverse Function Theorem are valid. For maps that are twice continuously differentiable, these estimates depend upon the…

We prove an implicit function theorem for non-commutative functions. We use this to show that if $p(X,Y)$ is a generic non-commuting polynomial in two variables, and $X$ is a generic matrix, then all solutions $Y$ of $p(X,Y)=0$ will commute…

In this paper, we present some implicit function theorems for set-valued mappings between Fr\'echet spaces. The proof relies on Lebesgue's Dominated Convergence Theorem and on Ekeland's variational principle. An application to the existence…

We prove an inverse function theorem of Nash-Moser type for maps between Fr\'echet spaces satisfying tame estimates. In contrast to earlier proofs, we do not use the Newton method, that is, we do not use quadratic convergence to overcome…

Derivative-based algorithms are ubiquitous in statistics, machine learning, and applied mathematics. Automatic differentiation offers an algorithmic way to efficiently evaluate these derivatives from computer programs that execute relevant…

Under general conditions, the equation $g(x,y) = 0$ implicitly defines $y$ locally as a function of $x$. In this article, we express divided differences of $y$ in terms of bivariate divided differences of $g$, generalizing a recent result…

In the context of the stream calculus, we present an Implicit Function Theorem (IFT) for polynomial systems, and discuss its relations with the classical IFT from calculus. In particular, we demonstrate the advantages of the stream IFT from…

We consider an equation of multiple variables in which a partial derivative does not vanish at a point. The implicit function theorem provides a local existence and uniqueness of the function for the equation. In this paper, we propose an…

We prove two versions of a global implicit function theorem, which involve no loss of derivative, for Keller's $ C_c^1 $-mappings between arbitrary Fr\'{e}chet spaces. Subsequently, within this framework, we apply these theorems to…

Under general conditions, the equation $g(x^1, ..., x^q, y) = 0$ implicitly defines $y$ locally as a function of $x^1, ..., x^q$. In this article, we express divided differences of $y$ in terms of divided differences of $g$, generalizing a…