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一点で微分可能であるが、それ以外の点で連続でない関数 [微分]

関数の定義域の一点で微分可能であるが、それ以外の定義域の点すべてで不連続な関数の一例。

  

このとき、f(x)は、x=0で微分可能で連続であるが、それ以外の点すべてで連続でない。

 

x=0で微分可能であることは、例えば、次のように証明されるだろう。

 

x≠0とする。

xが有理数のとき

  

xが無理数のとき

  

いずれにせよ、

  

よって、

  

x=0f(x)は微分可能なのだから、f(x)x=0で連続である。

 

ε-δ論法がよければ、(1)のところを次のようにすればいいだろう。

 

任意の正数ε>0に対してδ=ε>0にとれば、

  

 

x=0以外で連続でないことを証明するのは、例えば、次のようにすればいいだろう。

a≠0とする。

  

とすると、どんなδ>0をとっても

  

であるxが存在する(下図参照)。


doudakero.png

何故ならば、δ>0をどんなに小さくしても、aが有理数、xが無理数のとき、

  

であり、aが無理数、xが有理数のとき

  

となり、(2)を成立させるxが|x–a に存在するからである。

よって、a=0以外の全ての点でf(x)は連続ではなく、微分不可能である。

 

(※)

「関数f(x)が点aで連続である」のより正確な定義は、

  

したがって、点aで連続でないは、(2)を否定した

  

である。

 


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コメント 6

Josuedwepe

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by DavidRal (2021-08-09 00:55) 

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