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SVN
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For historical reasons the hyperbola always was considered to be one of the «classical» curves. The function, obviously, does not belong to C0. Apparently, is does not fit L2 or any other Lp? What is the smallest class?
Yes, you are right. I was thinking about interval [-1,1].Math_QED said:It depends.
$$(x \mapsto 1/x) \in C^0(\mathbb{R}\setminus \{0\})$$
In fact even
$$(x \mapsto 1/x) \in C^\infty(\mathbb{R}\setminus \{0\})$$
So I guess you will have to be a little more precise what your question is.
SVN said:Yes, you are right. I was thinking about interval [-1,1].
Math_QED said:So, how do you define x↦1/xx \mapsto 1/x in 00 then? You give it an arbitrary value?
Thank you!Infrared said:It is an element of ##L^p## when ##0<p<1## but these spaces aren't as nice; e.g. ##||f||_p=\left(\int |f|^p\right)^{1/p}## doesn't define a norm.
Well, of course, but I meant its classification from the viewpoint of functional analysis.mathwonk said:my first response would be to call it a rational function.
A function is a mathematical relationship between two quantities, where each input (or independent variable) has a unique output (or dependent variable).
The notation 1/x represents the reciprocal function, where the output is the inverse of the input. So, for example, if the input is 2, the output would be 1/2.
The domain of 1/x is all real numbers except for 0, since division by 0 is undefined. The range is also all real numbers except for 0, since there is no number that can be multiplied by 0 to get 1.
1/x is a discontinuous function, as it has a vertical asymptote at x=0. This means that the function is undefined at x=0 and has a break in its graph.
The reciprocal function has many applications in physics, engineering, and economics. For example, it can be used to model the relationship between force and distance in Hooke's Law, calculate the resistance of a circuit in electronics, and determine the optimal price of a product in economics.