A smooth function is a type of mathematical function that is continuously differentiable, meaning it has derivatives of all orders. This property allows for a smooth curve without any sharp corners or breaks. Smooth functions are important in calculus and analysis because they behave predictably, making them easier to work with in various applications, such as physics and engineering. Common examples of smooth functions include polynomials, sine, and cosine functions. These functions can be represented graphically with curves that flow seamlessly. The concept of smoothness is crucial in fields like differential geometry and optimization, where understanding the behavior of functions is essential for solving complex problems.