What does the term "pointwise" refer to?

Mr Davis 97

In reading mathematical texts (especially on functions), I come across the term "pointwise". There seems to be a paucity of information explaining it on the internet, so I turn to the forum for an intuitive explanation. The context is when the term is used as in "pointwise addition" or "pointwise multiplication" of functions.

SteamKing

Staff Emeritus
Homework Helper
In reading mathematical texts (especially on functions), I come across the term "pointwise". There seems to be a paucity of information explaining it on the internet, so I turn to the forum for an intuitive explanation. The context is when the term is used as in "pointwise addition" or "pointwise multiplication" of functions.
This often refers to a graphical method of adding or multiplying the ordinates of two curves to obtain a third curve, which is the sum or product of the first two curves.

Examine:

You take a series of points from curves ch.1 and ch.2 at the same x-location and add them together algebraically to make the bottom curve in the graphic above.

mathman

Let f,g,h be functions. f+g=h means f(x)+g(x)=h(x) for every x in the domain of these function. Similarly for any other operation involving functions. The term is more often used in the context of limits, where $f_n -> f$ can be defined in different ways, one of which is pointwise.

Mr Davis 97

So pointwise basically means that the way we manipulate functions is by considering their values at any x?

Mr Davis 97

Let f,g,h be functions. f+g=h means f(x)+g(x)=h(x) for every x in the domain of these function. Similarly for any other operation involving functions. The term is more often used in the context of limits, where $f_n -> f$ can be defined in different ways, one of which is pointwise.
Bump...

Mark44

Mentor
So pointwise basically means that the way we manipulate functions is by considering their values at any x?
No, at a specific x value.

"Pointwise" is commonly used in discussions about the convergence of sequences. See https://en.wikipedia.org/wiki/Pointwise_convergence for more info. One example given in this wiki article is the sequence $\{x^n\}$ on the interval [0, 1). This sequence converges pointwise to 0 for each x in [0, 1), but does not converge uniformly to 0 on the same interval.

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