This article aims at analyzing some topics related to the continuum and infinitesimals in the writings of Charles Sanders Peirce (1839-1914) and George Cantor (1845-1948). As for relevance, both Peirce and Cantor belong to the group of mathematicians that have studied the subject. The context can be taken into account from three different points of view: historical, philosophical, and mathematical. This study is restricted to the second half of the 19th century as well as the first two decades of the 20th century. On one hand, it is worth mentioning that Mathematics had developed significantly in Europe at that time. Regarding Cantor´s contributions, in Halle, Germany, the following can be highlighted: texts on real number (1872), disclosure of the Cantorian theory (1895-1897), formulation of the enumerable and continuum concepts (1874), the introduction of the concept of power (1878), study of the infinite, linear point sets (1879-1884). On the other hand, until 1880s, Science and Mathematics had remained underdeveloped in America with very few acknowledged contributions. Due to his findings regarding the set theory and mathematical logic, Peirce has stood out from his fellow European scientists. According to Belna (2011), Cantor spent his whole life explaining the continuum. It was first described in Grundlagen as a “perfect, one piece set” (Cantor, 1884, p. 208). Peirce became familiar with Cantor´s work around 1883-84 (CP 3.563). In the Collected Papers alone there are thirty-two passages in which Peirce discusses the Cantorian theory. There is also information of two letters sent to Cantor. Peirce gets to the conclusion that true continuum is different from any metric or orderly relation of components, it does not consist of real components (CP 3.631), it is general and could not be defined as a set, a collection of different components as Cantor´s definition (CP 4.640). According to Peirce, Cantor's definition of continuity is unsatisfactory as involving a vague reference to all the points (CP 6.125). Peirce´s continuum implies infinitesimals. Cantor was against infinitesimals, solution that was only properly acknowledged later on after the development of non-standard analysis. Therefore, despite the fact that Peirce and Cantor have shared the same object of study, their styles and the origin of their approaches and philosophical ideas are different.