Piecewise smooth function

A piecewise smooth function is a function defined on the set of real numbers , differentiable on each of the intervals that make up the domain of definition .

Formal Definition

Let given $x_{1}<x_{2}<\ldots <x_{n}$ ${\ displaystyle x_ {1} <x_ {2} <\ ldots <x_ {n}}$ - points of change of formulas.

Like all piecewise defined functions , a piecewise smooth function can be written on each of the intervals $(-\infty ;x_{1}),(x_{1};x_{2});\ldots (x_{n};+\infty )$ ${\ displaystyle (- \ infty; x_ {1}), (x_ {1}; x_ {2}); \ ldots (x_ {n}; + \ infty)}$ separate formula:

f(x)={\begin{cases}f_{0}(x),\quad x<x_{1}\\f_{1}(x),\quad x_{1}<x<x_{2}\\\cdots \\f_{n}(x),\quad x_{n}<x\end{cases}}

{\ displaystyle f (x) = {\ begin {cases} f_ {0} (x), \ quad x <x_ {1} \\ f_ {1} (x), \ quad x_ {1} <x <x_ {2} \\\ cdots \\ f_ {n} (x), \ quad x_ {n} <x \ end {cases}}}

Here $f_{i}(x)$ ${\ displaystyle f_ {i} (x)}$ - smooth functions .

If, moreover, the conditions for coordination are met

f_{i-1}(x_{i})=f_{i}(x_{i})=f(x_{i})

{\ displaystyle f_ {i-1} (x_ {i}) = f_ {i} (x_ {i}) = f (x_ {i})}

at

i=1,2,\ldots ,n

{\ displaystyle i = 1,2, \ ldots, n}

,

then the piecewise smooth function will be continuous . A continuous piecewise smooth function can serve as a spline .

Piecewise smooth function

Formal Definition

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