Through analyzing left and right-hand limits, we find that the function is continuous at the point. However, due to differing slopes from the left and right, the function is not differentiable at the edge point.
If a function is differentiable at x = c, then it is continuous at x = c. However, a function being continuous does not guarantee that it is differentiable, as demonstrated with an absolute value function.
Let's dive into examples of functions and their graphs, focusing on finding points where the function isn't differentiable. By examining various cases such as vertical tangents, discontinuities, and sharp turns, we gain a deeper understanding of the conditions that make a function non-differentiable.
Differentiability at a point confirms that a function must be continuous at that point. However, vice versa isn't true (You can have a continuous function with a sharp turn)
We examine a piecewise function to determine its continuity and differentiability at an edge point. By analyzing left and right hand limits, we establish continuity.
A function must be differentiable for the mean value theorem to apply. Learn why this is so, and how to make sure the theorem can be applied in the context of a problem.
We explore the connection between differentiability and continuity, demonstrating that if a function is differentiable at a point, it must also be continuous at that point. By examining the properties of limits and using algebra, we prove this relationship for a function at a given point.
