Let \(\mathbb{R}^n\) be a Euclidean space of dimension \(n\geq 2\). For a domain \(G\subset \mathbb{R}^n\), we denote by \(V_r(G)\) the set of functions \(f\in L_{\mathrm{loc}}(G)\) having zero integrals over all closed balls of radius r contained in G (if domain G does not contain such balls, we set \(V_r(G)=L_{\mathrm{loc}}(G)\)). Let E be a nonempty subset of \(\mathbb{R}^n\). In this paper we study the following questions related to the extension problem.

1) Which conditions guarantee the extension of a continuous function defined on E to a continuous function of class \(V_r(\mathbb{R}^n)\) defined on the whole \(\mathbb{R}^n\)?

2) If the above extension exists, obtain growth estimates of the extended function at infinity.

Theorem 1 of this paper shows that for a wide class of continuous functions on segment E defined in terms of the modulus of continuity, there exists an extension to a bounded function of class \((V_r\cap C)(\mathbb{R}^n)\) regardless of the length of segment E. A similar result is not true for open sets E with a diameter greater than 2r, even without conditions for extension growth. Theorem 1 also contains an estimate of the velocity decrease of the extended function at infinity in directions orthogonal to the segment E.

令\（\ mathbb {R} ^ n \）是维\（n \ geq 2 \）的欧几里得空间。对于域\（G \ subset \ mathbb {R} ^ n \），我们用\（V_r（G）\）表示函数\（f_in L _ {\ mathrm {loc}}（G）\中的集合）在包含在G中的所有半径为r的闭合球上具有零积分（如果域G不包含此类球，则设置 \（V_r（G）= L _ {\ mathrm {loc}}（G）\））。令E为\（\ mathbb {R} ^ n \）的非空子集。在本文中，我们研究以下与扩展问题有关的问题。

1）哪些条件可以保证将E上定义的连续函数扩展到整个\（\ mathbb {R} ^ n \）上定义的\（V_r（\ mathbb {R} ^ n）\）类连续函数？

2）如果存在上述扩展，请获取无穷大处扩展函数的增长估计。

本文的定理1表明，对于段E上根据连续模量定义的一系列连续函数，存在对\（（V_r \ cap C）（\ mathbb {R} ^ n）\）而不管段E的长度如何。即使没有扩展增长条件，直径大于2r的开放集合E也不存在类似的结果。定理1还包含在正交于线段E的方向上无穷大时扩展函数的速度下降的估计。