Continuous​ – This means that the limit of the function at a certain point is equal to the value of the function at that point.

Asymptote – These exist at x-values where a function has a denominator that's equal to zero with factors that can't be cancelled out. This is called a non-removable discontinuity (which shows up on a graph as an asymptote). In situations where the factors in the denominator can be cancelled out, this is called a removable discontinuity (which shows up on a graph as a hole).

Differentiable​ – This means that the derivative exists at that point (in other words, the limit definition of a derivative is the same from the left side as it is from the right side). In order to be differentiable, a function must be continuous​ at that point.