Hello everyone, and welcome back to another exciting lesson! Today, we embark on an intriguing journey of applying the binary search algorithm, which we have thoroughly covered in previous lessons, to continuous functions. This lesson aims to spark your curiosity and expand your understanding of the binary search algorithm. It will provide new insight on how to determine a specific function value within a continuous interval. This approach broadens the application of binary search from discrete space to continuous functions. So, let's unravel this exciting topic together!

Before we dive into binary search and continuous functions, let's refresh our understanding of what exactly continuous functions are. In the simplest terms, a function is a mapping from an input (or set of inputs) to an output. For instance, if we think about a Python function, it takes one or more arguments and returns an output based on the logic embedded within the function.

Continuous functions are those that produce a smooth, unbroken output for a continuous range of inputs without any abrupt changes or gaps. In mathematical terms, a function $f(x)$ is continuous at a point $x = a$ if the limit of as approaches from the left is equal to the limit of as approaches from the right, and these values are equal to . That means that: