Step 1: Understanding differentiability and continuity.
A function is said to be differentiable at a point if the derivative exists at that point. That is, if the limit
exists. If this limit exists, the function is differentiable at . For the function to be continuous at , the following condition must hold:
Step 2: Showing that differentiability implies continuity.
Assume that is differentiable at . By definition, we know that:
This is the definition of the derivative of at , which implies that the function has a well-defined rate of change at . Now, let's prove that is continuous at . For continuity, we need to show that:
From the definition of the derivative, we have:
where is the derivative of at . This means that as , the difference between and gets closer and closer to zero. Thus, by the definition of continuity:
proving that is continuous at .
Step 3: Conclusion.
Therefore, we have proven that if a function is differentiable at a point , then it must also be continuous at that point.