Displaying similar documents to “More on the Continuity of Real Functions”

Higher-Order Partial Differentiation

Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama (2012)

Formalized Mathematics

Similarity:

In this article, we shall extend the formalization of [10] to discuss higher-order partial differentiation of real valued functions. The linearity of this operator is also proved (refer to [10], [12] and [13] for partial differentiation).

Differential Equations on Functions from R into Real Banach Space

Keiko Narita, Noboru Endou, Yasunari Shidama (2013)

Formalized Mathematics

Similarity:

In this article, we describe the differential equations on functions from R into real Banach space. The descriptions are based on the article [20]. As preliminary to the proof of these theorems, we proved some properties of differentiable functions on real normed space. For the proof we referred to descriptions and theorems in the article [21] and the article [32]. And applying the theorems of Riemann integral introduced in the article [22], we proved the ordinary differential equations...

Riemann Integral of Functions from R into n -dimensional Real Normed Space

Keiichi Miyajima, Artur Korniłowicz, Yasunari Shidama (2012)

Formalized Mathematics

Similarity:

In this article, we define the Riemann integral on functions R into n-dimensional real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to the wider range. Our method refers to the [21].

The Differentiable Functions from R into R n

Keiko Narita, Artur Korniłowicz, Yasunari Shidama (2012)

Formalized Mathematics

Similarity:

In control engineering, differentiable partial functions from R into Rn play a very important role. In this article, we formalized basic properties of such functions.

The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

Keiko Narita, Noboru Endou, Yasunari Shidama (2013)

Formalized Mathematics

Similarity:

In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the...

The C k Space

Katuhiko Kanazashi, Hiroyuki Okazaki, Yasunari Shidama (2013)

Formalized Mathematics

Similarity:

In this article, we formalize continuous differentiability of realvalued functions on n-dimensional real normed linear spaces. Next, we give a definition of the Ck space according to [23].