A nowhere differentiable continuous function is a mathematical function that is continuous at every point, but does not have a derivative at any point. This means that the function is not smooth and has sharp corners or breaks at every point.
Yes, a nowhere differentiable continuous function can exist. In fact, there are many examples of such functions, such as the famous Weierstrass function, which is continuous everywhere but differentiable nowhere.
Nowhere differentiable continuous functions are important in mathematics because they challenge our understanding of continuity and differentiability. They also have applications in areas such as fractal geometry and signal processing.
The main difference between nowhere differentiable continuous functions and regular continuous functions is that the former does not have a derivative at any point, while the latter has a derivative at most points. This means that nowhere differentiable functions are much more irregular and unpredictable.
While nowhere differentiable continuous functions may seem abstract, there are actually many real-world examples of them. For instance, the coastline of a country or continent can be modeled as a nowhere differentiable continuous function, as it has many sharp turns and irregularities at different scales.