Pointwise combination of functions: Difference between revisions

Latest revision as of 15:36, 24 September 2021

Definition

For a binary operation

Suppose we have a binary operation (such as addition, subtraction, multiplication, division) on real numbers. A pointwise combination of two functions that take real values is a new function that first applies the two functions individually and then applies the binary operation.

For instance, pointwise sum of functions $f$ and $g$ is the function $x\mapsto f(x)+g(x)$ . Similarly, pointwise product of functions $f$ and $g$ is thee function $x\mapsto f(x)g(x)$ .

The term pointwise is an allusion to the inputs to the function as points (thinking of the function geometrically). Pointwise refers to the idea that for each point, the values of the functions at that point are combined.

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 Suppose we have a binary operation (such as addition, subtraction, multiplication, division) on real numbers. A '''pointwise combination''' of two functions that take real values is a new function that first applies the two functions individually and then applies the binary operation.
-For instance, [[pointwise addition of functions]] <math>f</math> and <math>g</math> is the function <math>x \mapsto f(x) + g(x)</math>. Similarly, [[pointwise multiplication of functions]] <math>f</math> and <math>g</math> is thee function <math>x \mapsto f(x)g(x)</math>.
+For instance, [[pointwise sum of functions]] <math>f</math> and <math>g</math> is the function <math>x \mapsto f(x) + g(x)</math>. Similarly, [[pointwise product of functions]] <math>f</math> and <math>g</math> is thee function <math>x \mapsto f(x)g(x)</math>.
 The term '''pointwise''' is an allusion to the inputs to the function as points (thinking of the function geometrically). ''Pointwise'' refers to the idea that ''for each point, the values of the functions at that point are combined.''