Abstract
In the early 19th century, mathematicians believed that a continuous function is always differentiable except at certain points. However, further exploration between the relationship of continuity and differentiability led to the discovery of continuous nowhere differentiable functions (also known as pathological functions due to their controversial nature). Karl Weierstrass was famously attributed to this field due to being the first to publicly present his work before the Berlin Academy in 1872. His pathological function shocked the mathematical community and was known to spark debates around the question of whether continuous functions were necessarily nowhere differentiable. After the publication of his work, many others contributed to this area of analysis. This research study explores the historical background of continuous nowhere differentiable functions through the analysis of continuity and differentiability in infinite series. Then it examines the properties of these functions and other variations of continuous nowhere differentiable functions.
