Takagi's Theorem is a result in the field of mathematics, specifically in the area of real analysis. It states that every continuous function defined on a closed interval can be uniformly approximated by a sequence of continuous functions that are piecewise linear. This means that for any continuous function, you can find simpler functions that get very close to it, making it easier to analyze and work with. The theorem is named after the Japanese mathematician Takahashi Takagi, who introduced this concept in the early 20th century. It highlights the importance of approximation in mathematics, showing how complex functions can be represented in simpler forms while retaining their essential properties.