We divided this thesis into three larger chapters in which we observed some elementary aspects of continuity. In the first chapter, we defined continuous functions and studied sequences. Then we considered bounded subsets of R and some properties of sequences, and we also introduced the notion of the limit of a function and examined its connection with continuity. At the end of the first chapter, we stated the axiom of completeness of real numbers. In the second chapter we dealt with continuous functions on a segment. We proved that a continuous function on a segment takes on all intermediate values and introduced the notions of an accumulation point of a sequence. We also proved that every bounded sequence of real numbers has a convergent subsequence and that every continuous function on a segment is bounded. In the third chapter we studied continuity in the context of metric spaces. At the beginning, we stated the definition of the metric space and some basic examples, and then we dealt with open sets in the metric space. At the end, we observed continuous functions between metric spaces.