Editing Derivative (section)

== Continuity and differentiability ==
{{multiple image
| total_width       = 480
| image1            = Right-continuous.svg
| caption1          = This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a [[jump discontinuity]]).
| image2            = Absolute value.svg
| caption2          = The absolute value function is continuous but fails to be differentiable at {{math|''x'' {{=}} 0}} since the tangent slopes do not approach the same value from the left as they do from the right.
}}

If <math> f </math> is [[differentiable]] at {{tmath|1= a }}, then <math> f </math> must also be [[continuous function|continuous]] at <math> a </math>.{{sfnm|1a1=Gonick|1y=2012|1p=156 |2a1=Thomas et al.|2y=2014|2p=114 |3a1=Strang et al. |3y=2023 |3p=[https://openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function 237] }} As an example, choose a point <math> a </math> and let <math> f </math> be the [[step function]] that returns the value 1 for all <math> x </math> less than {{tmath|1= a }}, and returns a different value 10 for all <math> x </math> greater than or equal to <math> a </math>.  The function <math> f </math> cannot have a derivative at <math> a </math>. If <math> h </math> is negative, then <math> a + h </math> is on the low part of the step, so the secant line from <math> a </math> to <math> a + h </math> is very steep; as <math> h </math> tends to zero, the slope tends to infinity. If <math> h </math> is positive, then <math> a + h </math> is on the high part of the step, so the secant line from <math> a </math> to <math> a + h </math> 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 <math> f(x) = |x| </math> is continuous at {{tmath|1= x = 0 }}, but it is not differentiable there. If <math> h </math> is positive, then the slope of the secant line from 0 to <math> h </math> is one; if <math> h </math> is negative, then the slope of the secant line from <math> 0 </math> to <math> h </math> is {{tmath|1= -1 }}.{{sfnm|1a1=Gonick|1y=2012|1p=149 |2a1=Thomas et al.|2y=2014|2p=113 |3a1=Strang et al. |3y=2023 |3p=[https://openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function 237]}}  This can be seen graphically as a "kink" or a "cusp" in the graph at <math>x=0</math>. Even a function with a smooth graph is not differentiable at a point where its [[Vertical tangent|tangent is vertical]]: For instance, the function given by <math> f(x) = x^{1/3} </math> is not differentiable at <math> x = 0 </math>. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.{{sfnm|1a1=Gonick|1y=2012|1p=156 |2a1=Thomas et al.|2y=2014|2p=114 |3a1=Strang et al. |3y=2023 |3pp=[https://openstax.org/books/calculus-volume-1/pages/3-2-the-derivative-as-a-function 237–238]}}

Most functions that occur in practice have derivatives at all points or [[Almost everywhere|almost every]] point. Early in the [[history of calculus]], many mathematicians assumed that a continuous function was differentiable at most points.{{sfnm
 | 1a1 = Jašek | 1y = 1922
 | 2a1 = Jarník | 2y = 1922
 | 3a1 = Rychlík | 3y = 1923
}} Under mild conditions (for example, if the function is a [[monotone function|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]].{{sfn|David|2018}} 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.<ref>{{harvnb|Banach|1931}}, cited in {{harvnb|Hewitt|Stromberg|1965}}.</ref>