what does it mean to be differentiable

In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, a function is differentiable if it has a non-vertical tangent line at each interior point in its domain. If x0 is an interior point in the domain of a function f, then f is said to be differentiable at x0 if the derivative exists. In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). A function is said to be differentiable on U if it is differentiable at every point of U. A continuous function is not necessarily differentiable, but a differentiable function is necessarily continuous (at every point where it is differentiable). To determine differentiability, one can use limits and continuity. The definition of differentiability is expressed as follows: f is differentiable on an open interval (a,b) if the limit of the difference quotient exists for every c in (a,b). If a function is differentiable, then its derivative must be continuous as well.