Is there any concept of continuity in p-adic field (field of p-adic numbers) ?
Is there concept of continuous curve and surfaces in p-adic field?
Answer:
I know that the field of p-adic numbers is totally disconnected.
So question of continuous curves and surface or cncept of continuity , differentiability does not arise.
But I am not getting confidence .
Can someone give me more details about the above questions?
Thanks in advance
The $p$-adic numbers are a metric space, and thus a topological space. Therefore continuity is well defined, either by the $\varepsilon,\delta$ definition that you know from Calculus, or by the simpler-looking criterion that if $X$ and $Y$ are topological spaces, then $f:X\to Y$ is continuous if and only if for every open subset $U\subset Y$, the set $f^{-1}(U)$ is an open subset of $X$.
To define a curve in the $p$-adic concept, you have to know what you’re looking for, know what the definition ought to be. You may look at a $2$-variable polynomial $F(X,Y)$ with $\Bbb Q_p$-coefficients, and ask for all $(a,b)\in\Bbb Q_p\times\Bbb Q_p$ such that $F(a,b)=0$. This is the outlook (suitably generalized and jazzed up) that is taken in Algebraic Geometry. Just as, for instance $Y^2-X^2(X-1)^2$ gives you a nice curve in $\Bbb R\times\Bbb R$, so the same polynomial, having integer coefficients, makes perfectly good $p$-adic sense, though I don’t think that anyone would say that the $p$-adic curve looked like a butterfly.
Or you may think of a curve as parametrized: in the familiar real context, you might take a continuous map of the closed unit interval into $\Bbb R^2$ and ask for the form of the image (“range”) of this map. Similarly, taking the analog of the unit interval to be the subset $\Bbb Z_p$ of $p$-adic numbers within $\Bbb Q_p$, you might ask for continuous maps of $\Bbb Z_p$ into $\Bbb Q_p^2$.
What you ought never do, is try for continuous maps of $[0,1]$ into $\Bbb Q_p^2$: as you know, the latter is totally disconnected, so that the only maps to it from a connected set like $[0,1]$ will be constant.
In the $p$-adic world, it is perfectly possible to define differentiability and ask about the derivative of a function. Total disconnectedness offers no impediment whatever. Notice, by the way, that although there are no interesting continuous maps from the reals to the $p$-adics, there are plenty in the opposite direction. Indeed, the $p$-adic absolute-value function is just one such.
What’s the moral of my sermon? That there are far more interesting analytic and geometric phenomena in the $p$-adic world than you assumed. It is true, however, that the flavor of these is very different from what you know from the Archimedean world.