Update

function.md

@@ -4,7 +4,7 @@ Function is a very basic term in [mathematics](math.md) and [programming](progra
 
 ## Mathematical Functions
 
-In mathematics functions can be defined and viewed from different angles but a function is basically anything that assigns each member of some set *A* (so called *domain*) exactly one member of a potentially different set *B* (so called *codomain*). A typical example of a function is an equation that from one "input number" computes another number, for example:
+In mathematics functions can be defined and viewed from different angles, but it is essentially anything that assigns each member of some [set](set.md) *A* (so called *domain*) exactly one member of a potentially different set *B* (so called *codomain*). A typical example of a function is an equation that from one "input number" computes another number, for example:
 
 *f(x) = x / 2*
 
@@ -50,9 +50,11 @@ Plotting functions of multiple parameters is more difficult because we need more
 
 Functions can have certain properties such as:
 
-- being **[bijective](bijection.md)**: A bijective function pairs exactly one element from the domain with one element from codomain and vice versa, i.e. for every result (element of codomain) of the function it is possible to unambiguously say which input created it. For bijective functions we can create **[inverse functions](inverse_function.md)** that reverse the mapping (e.g. [arcus sine](asin.md) is the inverse of a [sin](sin.md) function that's limited to the interval where it is bijective). For example *f(x) = 2 * x* is bijective with its inverse function being *f^(-1)(x) = x / 2*, but *f2(x) = x^2* is not bijective because e.g. both 1 and -1 give the result of 1. 
+- being **[bijective](bijection.md)**: Pairs exactly one element from the domain with one element from codomain and vice versa, i.e. for every result (element of codomain) of the function it is possible to unambiguously say which input created it. For bijective functions we can create **[inverse functions](inverse_function.md)** that reverse the mapping (e.g. [arcus sine](asin.md) is the inverse of a [sin](sin.md) function that's limited to the interval where it is bijective). For example *f(x) = 2 * x* is bijective with its inverse function being *f^(-1)(x) = x / 2*, but *f2(x) = x^2* is not bijective because e.g. both 1 and -1 give the result of 1. 
 - being an **[even function](even_function.md)**: For this function it holds that *f(x) = f(-x)*, i.e. the plotted function is symmetric by the vertical axis. Example is the [cosine](cos.md) function.
 - being an **[odd function](odd_function.md)**: For this function it holds that *-f(x) = f(-x)*, i.e. the plotted function is symmetric by the center point [0,0]. Example is the [sine](sin.md) function.
+- being **[differentiable](differentiable.md)**: Its [derivative](derivative.md) is defined everywhere.
+- ...
 
 In context of functions we may encounter the term [composition](composition.md) which simply means chaining the functions. E.g. the composition of functions *f(x)* and *g(x)* is written as *(f o g)(x)* which is the same as *f(g(x))*.
 
@@ -79,7 +81,7 @@ Functions commonly used in mathematics range from the trivial ones (such as the
 
 ## Programming Functions
 
-In programming the definition of a function is less strict, even though some languages, namely [functional](functional.md) ones, are built around purely mathematical functions -- for distinction we call these strictly mathematical functions **pure**. In traditional languages functions may or may not be pure, a function here normally means a **subprogram** which can take parameters and return a value, just as a mathematical function, but it can further break some of the rules of mathematical functions -- for example it may have so called **[side effects](side_effect.md)**, i.e. performing additional actions besides just returning a number (such as modifying data in memory, printing something to the screen etc.), or use randomness and internal states, i.e. potentially returning different numbers when invoked (called) multiple times with exactly the same parameters. These functions are called **impure**; in programming a *function* without an adjective is implicitly expected  to be impure. Thanks to allowing side effects these functions don't have to actually return any value, their purpose may be to just invoke some behavior such as writing something to the screen, initializing some hardware etc. The following piece of code demonstrates this in [C](c.md):
+In programming the definition of a function is less strict, even though some languages, namely [functional](functional.md) ones, are built around purely mathematical functions -- for distinction we call these strictly mathematical functions **pure**. In traditional languages functions may or may not be pure, a function here normally means a **subprogram** which can take parameters and return a value, just as a mathematical function, but it can further break some of the rules of mathematical functions -- for example it may have so called **[side effects](side_effect.md)**, i.e. performing additional actions besides just returning a number (such as modifying data in memory which can be read by others, printing something to the screen etc.), or use randomness and internal states, i.e. potentially returning different numbers when invoked (called) multiple times with exactly the same arguments. These functions are called **impure**; in programming a *function* without an adjective is implicitly expected  to be impure. Thanks to allowing side effects these functions don't have to actually return any value, their purpose may be to just invoke some behavior such as writing something to the screen, initializing some hardware etc. The following piece of code demonstrates this in [C](c.md):
 
 ```
 int max(int a, int b, int c) // pure function