**Understanding the Concept of Differentiability in Mathematics**

**What Does it Mean When f is Differentiable at x_0?**

**When we talk about a function f being differentiable at a point x_0**, we are essentially referring to the smoothness of the function at that specific point. **In mathematical terms, differentiability at x_0 means that the function f has a well-defined derivative at x_0.** This concept is crucial in calculus and plays a significant role in understanding the behavior of functions at specific points.

**So, what exactly does it mean for a function to be differentiable at a point x_0?** Let’s break it down into simpler terms. When a function is differentiable at x_0, it indicates that the function is continuous at x_0 and also has a well-defined tangent at that point. In other words, the function exhibits a smooth and continuous behavior without any abrupt changes or corners at x_0.

**Continuity:**A function must be continuous at x_0 for it to be differentiable at that point.**Tangent Line:**The existence of a well-defined tangent line at x_0 is a key characteristic of differentiability.**Smoothness:**Differentiability implies that the function behaves smoothly at x_0 without any sudden jumps.

**It’s important to note that not all functions are differentiable at every point.** Some functions may have points where they are not differentiable, such as sharp corners or discontinuities. Understanding the concept of differentiability helps us analyze the behavior of functions and make predictions about their characteristics at specific points.

**So, in summary, when we say that a function f is differentiable at a point x_0, we are essentially stating that the function is continuous, smooth, and has a well-defined tangent at that particular point.** This concept forms the foundation of calculus and is essential for studying the behavior of functions in mathematics.