Proving Differentiability Implies Continuity

Prove that if a function is differentiable at some point $C$, then it is also continuous at that point $C$.

In this problem, you are tasked with demonstrating a fundamental concept in real analysis: the relationship between differentiability and continuity. Understanding this connection is crucial, as it lays the groundwork for further exploration of more complex topics in analysis. When a function is differentiable at a point, it means that the function has a well-defined tangent at that point, implying that it behaves predictably when closely examined around that point. This notion of 'smoothness' is a key aspect of differentiability and is intrinsically linked to continuity. Continuity ensures that a function does not have any jumps or holes at the point of interest, making it a prerequisite for differentiability. The challenge in this problem is to provide a logical argument that starts with the assumption of differentiability and follows through to establish continuity. By doing this, you will learn how to construct proofs that require linking definitions and theorems in a cohesive manner, an essential skill in real analysis. Furthermore, you will deepen your understanding of how local properties of a function can influence its overall behavior at specific points, preparing you for more advanced studies in the subject.