Uniform Continuity of Fractal Interpolation Function

In order to research analysis properties of fractal interpolation function generated by the iterated function system defined by affine transformation, the continuity of fractal interpolation function is proved by the continuous definition of function and the uniform continuity of fractal interpolation function is proved by the definition of uniform continuity and compactness theorem of sequence of numbers or finite covering theorem in this paper.-e result shows that the fractal interpolation function is uniformly continuous in a closed interval which is from the abscissa of the first interpolation point to that of the last one.


Introduction
In 1960s, fractal geometry was regarded as a new interdiscipline firstly discovered by American mathematician Mandlebrot [1][2][3][4][5]. On the one hand, because most extremely irregular graphics in nature and very irregular social phenomenon are researched in the fractal geometry field, fractal geometry is called natural geometry. erefore, fractal geometry is applied in almost all fields, such as mathematics, physics, chemistry, engineering, social science, and art [6][7][8][9]. On the other hand, for research on the natural properties of fractal itself, many methods used in researching fractals have been found by experts from 1960s to now, for example, fractal dimension method [10], multifractal spectrum method [11], adaptive fuzzy output-feedback method of nonlinear system [12], and nonlinear iterated method [13]. Especially, the fractal interpolation function method has been paid more and more attention by mathematicians. e theory of fractal interpolation function generated by the iterated function system defined by affine transformation was firstly proposed by Barnsley [14][15][16] and Massopust [17,18]. ey found that any part of a fractal graphic is similar to the whole, so they used mathematical language to express the similar iterated process. at is to say, first, the iterated function system consisting of affine transformation is defined and it is proved that the iterated function system has a unique attractor that is the fixed point. Second, according to the theory of iterated function system and selfsimilar theory, complex fractal graphic, the graphic of fractal interpolation curve (refer with Figure 1), or fractal interpolation surface (refer with Figure 2) can be generated by computer program. Finally, the dimension theory and integrability of fractal interpolation function have been studied by Barnsley and Massopust. Based on the research of fractal interpolation function above, the continuity and uniform continuity of the fractal interpolation function-generated iterated function systemdefined affine mapping are proved in the paper.

Main Concepts and Lemmas
Definition 1 (see [19,20]). Let f be a function defined on interval I. If ∀x ∈ I and ∀ε > 0, there is a δ > 0, so that for any x ∈ I and |x − x| < δ ⟹ |f(x) − f(x)| < ε, we call that f is continuous on the point x and the f is called continuous function on I.
Definition 5 (see [14]). Let (x i , y i ) ∈ R 2 ; i � 0, 1, 2, . . . , n be a set of points, where x 0 < x 1 < x 2 < · · · < x n . An interpolation function corresponding to this set of data is a function f: (1) e points (x i , y i ) are called the interpolation points. It is called that the function of f interpolates the data and that the graph of f passes through the interpolation points.
Lemma 1 (see [19,20]). If a sequence x n ∞ n�1 is bounded, the sequence x n has a convergent subsequence x n k ∞ k�1 .
Lemma 2 (see [19,20] Lemma 3 (see [14][15][16][17][18]). Let n be a positive integer greater than 1. Let R 2 ; w i , i � 1, 2, . . . , n denote the IFS defined above, associated with the data set Let the vertical scaling factor d i obey 0 ≤ d i < 1 for i � 1, 2, . . . , n. en, there is a metric d on R 2 , equivalent to the Euclidean metric, such that the IFS is hyperbolic with respect to d. In particular, there is a unique nonempty compact set G ⊂ R 2 , such that In particular, an IFS of the form R 2 ; w i , i � 1, 2, . . . , n is considered, where the mapping is an affine transformation of the special structure e transformations are constrained by the data according to a i , e i , c i , and f i can be solved from equations (4)-(5) in terms of the data and vertical scaling factor d i : Lemma 4 (see [14][15][16]   and f(x n ) � y n . e metric is defined by the following formula: (10) en, (F, d) is a complete metric space. Let the real numbers a i , c i , e i , and f i , be defined by equations (5)- (9). Define a mapping T: F ⟶ F by where en, Tf is continuous on the interval [x i−1 , x i ] and T is a contraction mapping on (F, d), so T possesses a unique fixed point in F. at is, there exists a function f ∈ F such that Tf � f, ∀f ∈ F. (14) e function f is called fractal interpolation function. e abbreviation "FIF" is used for "fractal interpolation function."

Theorem 1. Let function f be a FIF generated by IFS mentioned in Definition 4 and defined by affine mapping referred from equations (3)-(14) and Lemmas 3 and 4. en, the FIF is continuous on the closed interval on the closed interval
Proof. ∀x ∈ [x 0 , x n ], according to Lemmas 3 and 4 and equations (11) to (14), ∀ε > 0, and Select en, we select so ∀ε > 0, for any x ∈ [x 0 , x n ] and |x − x| < δ, such that Proof.
e contradiction method will be used in the proof. Suppose that FIF is not uniformly continuous on the closed interval [x 0 , x n ]. According to Definition 2, so that Mathematical Problems in Engineering 3 that is, but Because FIF is a continuous function on the closed interval [x 0 , x n ], (24) From the relationship between limit of function and that of number sequence, that is, which is contrary to ε 0 > 0. In other words, at first, the contrary hypothesis to the conclusion of eorem 2 makes ε 0 > 0 and ε 0 ≤ 0 contradictory. So, f is uniform continuous function on the interval [x 0 , x n ].
For the proof of eorem 2, we have the following second method to prove eorem 2.