Derivative
Continuity and differentiability

If f {\displaystyle f} is differentiable at a {\displaystyle a}, then f {\displaystyle f} must also be continuous at a {\displaystyle a}. As an example, choose a point a {\displaystyle a} and let f {\displaystyle f} be the step function that returns the value 1 for all x {\displaystyle x} less than a {\displaystyle a}, and returns a different value 10 for all x {\displaystyle x} greater than or equal to a {\displaystyle a}. The function f {\displaystyle f} cannot have a derivative at a {\displaystyle a}. If h {\displaystyle h} is negative, then a + h {\displaystyle a+h} is on the low part of the step, so the secant line from a {\displaystyle a} to a + h {\displaystyle a+h} is very steep; as h {\displaystyle h} tends to zero, the slope tends to infinity. If h {\displaystyle h} is positive, then a + h {\displaystyle a+h} is on the high part of the step, so the secant line from a {\displaystyle a} to a + h {\displaystyle a+h} has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function given by f (x) = | x | {\displaystyle f(x)=|x|} is continuous at x = 0 {\displaystyle x=0}, but it is not differentiable there. If h {\displaystyle h} is positive, then the slope of the secant line from 0 to h {\displaystyle h} is one; if h {\displaystyle h} is negative, then the slope of the secant line from 0 {\displaystyle 0} to h {\displaystyle h} is − 1 {\displaystyle -1}. This can be seen graphically as a "kink" or a "cusp" in the graph at x = 0 {\displaystyle x=0}. Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by f (x) = x 1 / 3 {\displaystyle f(x)=x^{1/3}} is not differentiable at x = 0 {\displaystyle x=0}. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.

Most functions that occur in practice have derivatives at all points or almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions (for example, if the function is a monotone or a Lipschitz function), this is true. However, in 1872, Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function. In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions. Informally, this means that hardly any random continuous functions have a derivative at even one point.
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