Continuous function between functions

Question

Is there a notion of a "continuous functions between continuous functions" over some interval $I\in \mathbb{R}$?

For instance, some $G_t$ that satisfies $G_0=f(x), G_1=g(x)$, which varies continuously between the $f(x)$ and $g(x)$ with $t\in[0,1]$.

For this to make sense, I can imagine we would require $J_a(t)=G_t(a)$ to be a continuous function over $t\in [0,1]$. This would then need to apply for all $a \in I$.

Are you looking for the word "homotopy" ? https://en.wikipedia.org/wiki/Homotopy — Donald Splutterwit, Oct 10 '20 at 15:32
@DonaldSplutterwit I think that is the word I am looking for. Thanks! — Kcronix, Oct 10 '20 at 15:38
@Jean-ClaudeArbaut Thanks for the example! That is essentially the same as the expression for any point between two points in 2D or 3D (or infinite dimensional vector spaces, as are functions). — Kcronix, Oct 10 '20 at 15:41

score 1 · Accepted Answer · answered Oct 10 '20 at 15:32

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https://en.wikipedia.org/wiki/Homotopy#Formal_definition

You might have a look at the concept of "Homotopy".

answered Oct 10 '20 at 15:32

Lilla

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score 1 · Answer 2 · answered Oct 10 '20 at 15:34

I think that homotopy maybe answers your question. https://en.wikipedia.org/wiki/Homotopy

So the definition is: A homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined to be a continuous function ${\displaystyle H:X\times [0,1]\to Y}$ from the product of the space $X$ with the unit interval $[0,1]$ to $Y$ such that ${\displaystyle H(x,0)=f(x)}$ and ${\displaystyle H(x,1)=g(x)}$ for all ${\displaystyle x\in X}$.

score 1 · Answer 3 · answered Oct 10 '20 at 15:34

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A convex combination of two continuous functions is continuous as well. So that is one example.

$$G_t(x) = (1-t)G_0(x) + tG_1(x)$$

answered Oct 10 '20 at 15:34

Annika

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Continuous function between functions

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