File:PiecewiseMonoDiff.pdf
Summary
| Description |
English: The function is an example of a piecewise monotonic and piecewise differentiable function. The intervals the domain is partioned in differ by property, they are and for monotonicity, and , , and for differentiability,
The function definition doesn't involve an explicit case distinction. While the usual definition uses a case distinction, so does addition on natural numbers which is employed, directly or indirectly, by all arithmetic operations on , , , , and . Hence, indirect use of case distinctions cannot be avoided in any function definition whatsoever. The function (not depicted) is piecewise monotonic but built from arithmetic operations only. |
| Date | |
| Source | Own work |
| Author | Jochen Burghardt |
| Other versions | File:PiecewiseMonoDiff.pdf * File:PiecewiseMonoDiff_svg.svg |
![{\displaystyle (-\infty ,0]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0241c015ef4c611c9c9aeafb395e7c4a16178405)
![{\displaystyle (-\infty ,-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3159526ce3102ba58727dfc08cb732b19fdd0b96)
![{\displaystyle [-1,+1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daa72f1a806823ec94fda7922597b19cbda684f4)
Gnuplot source code |
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set title "min ( 1, x^2 ) is piecewise monotonic and piecewise differentiable"
set grid
#set key left bottom
set xrange [-4:+4]
set yrange [-0.25:+1.10]
set samples 1000
f(x) = ( x*x < 1 ? x*x : 1 )
plot f(x) with lines lw 2 lt 7 title ""
set label "min ( 1, x^2 )" at -3.70 , 0.95 tc lt 7
# mono
set arrow from 0.00 , -0.10 to 0.00 , -0.06 nohead lc 1 lw 2
set arrow from -0.05 , -0.08 to -4.00 , -0.08 lc 1 lw 1
set arrow from +0.05 , -0.08 to +4.00 , -0.08 lc 1 lw 1
set label "monotonicity intervals" at -3.70 , -0.05 tc lt 1
# diff
set arrow from -1.00 , -0.20 to -1.00 , -0.16 nohead lc 2 lw 2
set arrow from +1.00 , -0.20 to +1.00 , -0.16 nohead lc 2 lw 2
set arrow from -1.05, -0.18 to -4.00 , -0.18 lc 2 lw 1
set arrow from -0.95, -0.18 to +0.95 , -0.18 nohead lc 2 lw 1
set arrow from +1.05, -0.18 to +4.00 , -0.18 lc 2 lw 1
set label "differentiability intervals" at -3.70 , -0.15 tc lt 2
pause -1
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Licensing
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