Why not differentiable at cusp

Hence, differentiability is when the slope of the tangent line equals the limit of the function at a given point. This directly suggests that for a function to be differentiable, it must be continuous, and its derivative must be continuous as well. Now, this leads us to some very important implications — all differentiable functions must therefore be continuous, but not all continuous functions are differentiable!

Simply put, differentiable means the derivative exists at every point in its domain. Consequently, the only way for the derivative to exist is if the function also exists i. Thus, a differentiable function is also a continuous function. We can easily observe that the absolute value graph is continuous as we can draw the graph without picking up your pencil. I can show that it's not an immersed submanifold, but there was something unsatisfying about it. That's kinda satisfying, I guess.

Here we talk about an extra level of smoothness i. It is a matter of what "reminds me of a line" means. Add a comment. Ted Shifrin Ted Shifrin Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post.

Linked 9. Related 2. Hot Network Questions. Question feed. Ben Davis July 7, How do you know if a limit is one sided? Do one-sided limits always exist? Can a graph be continuous at a corner?

Why do derivatives not exist at sharp corners? Can derivatives be zero? Are endpoints critical points? What is the difference between a corner and a cusp? What does it mean when the tangent line is vertical? Why are cusps and corners not differentiable? Are corners and cusps differentiable? Do limits exist at cusps? Are functions differentiable at sharp corners? Can a function be differentiable at a hole?

What kinds of functions are not differentiable? Why are vertical tangents not differentiable? How do you know if a tangent line is vertical? How do you know if a function is not differentiable?