From Math Images
Can you say that is an everywhere differentiable function? I feel like that a subtlety missing in this page. Also I'd retitle the "basic description" section to be something else, for the same reasons in Alan's comment below.
I think you have a typo in the final term for the tangent line section you have an h were it should be an a.
I think your basic information needs to be revised. If a hypothetical reader has no calculus backgrounds; the basic information would be over his or her head. The limit definition is very mathematical and could be off putting to someone not familiar with it.
Saying something like "A point is differentiable if you can draw a tangent at this point...." and then have some tangent line pictures would probably be a mathematically gentler introduction.
Your basic information should perhaps go into the "a more mathematical description" part of the page.
Also I think it would better to have the page in the helper page format (with no main image). The parabola seems to be something ubiquitous in middle and high school curriculums, so I don't think people will be attracted to a picture of it.
Everything else looks good though.
In the first example, instead of saying that the function "skips" that point, have you considered using the term undefined? If you're trying to avoid really mathematical language, maybe say "behaves differently".
Also, I don't feel like it's clear that you are taking two separate one sided limited to come up with that and . Have you considered writing out the two limits separately?
Are you planning on adding a couple more sentences with examples of functions that are continuous but not differentiable, or at least pointing back to your corner function?