In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be differentiable at if the derivative exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be continuously differentiable if its derivative is also a continuous function over the domain of the function . Generally speaking, is said to be of class if its first derivatives exist and are continuous over the domain of the function . Differentiability of real functions of one variable A function , defined on an open set , is said to be differentiable at if the derivative exists. This implies that the function is continuous at . This function is said to be differentiable on if it is differentiable at every point of . In this case, the derivative of is thus a function from into A continuous function is not necessarily differentiable, but a differentiable function is necessarily continuous (at every point where it is differentiable) as being shown below (in the section Differentiability and continuity). A function is said to be continuously differentiable if its derivative is also a continuous function; there exists a function that is differentiable but not continuously differentiable as being shown below (in the section Differentiability classes). Differentiability and continuity If is differentiable at a point , then must also be continuous at . In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly. Most functions that occur in practice have derivatives at all points or at almost every point. However, a result of Stefan Banach states that the set of functions that have a derivative at some