Can someone confirm if my current definitions of continuity and differentiability are correct?

So continuity means that our point:

A) Is defined

B) The limit on the right and left side of the point equal the y value of our point

Differentiability means the derivative at the point but a little to the left equals the derivative of the point but a little to the right. So for example, for a point to be differentiable at x = 0, the derivatives at x = 0 but a little less and the derivative at x = 0 but a little more should be equal.

Any mistakes in my understanding? My brain hurts trying to understand the definitions