ﻣﺸﺘﻖ ﺗﺎﺑﻊ، ﺗﻌﻤﯿﻢ ﺗﻌﺮﯾﻒ ﮐﺎراﺗﺌﻮدري

Derivative of a function, Generalization of the Caratheodory Definition

ﺗﻌﺮﯾﻒ ﻣﺘﺪاول ﻣﺸﺘﻖ، ﻣﺠﻤﻮﻋﮥ ﺗﺎﺑﻊﻫﺎي ﻣﺸﺘﻖﭘﺬﯾﺮ را ﺑﺴﯿﺎر ﮐﻮﭼﮏﺗﺮ از ﻣﺠﻤﻮﻋﮥ ﺗﺎﺑﻊﻫﺎي ﭘﯿﻮﺳﺘﻪ ﻣﯽﺳﺎزد. ﺑﺴﯿﺎري از ﺗﺎﺑﻊﻫﺎي ﯾﮏ ﻣﺘﻐﯿﺮي ﺑﺎ ﺗﻌﺮﯾﻒ ﻣﻮﺟﻮد ﻣﺸﺘﻖﭘﺬﯾﺮ ﻧﯿﺴﺘﻨﺪ و ﺑﺮرﺳﯽ ﺗﻐﯿﯿﺮات آﻧﻬﺎ ﺑﻪ ﮐﻤﮏ ﺗﻌﺮﯾﻒ ﻣﻮﺟﻮد ﻣﺸﺘﻖ، ﻣﯿﺴﺮ ﻧﯿﺴﺖ. در اﯾﻦ ﻣﻘﺎﻟﻪ، ﺑﺎ اﺳﺘﻔﺎده از ﺗﻌﺮﯾﻒ ﮐﺎراﺗﺌﻮدري، اﺑﺘﺪا ﺑﻪ اراﺋﮥ ﺗﻌﺮﯾﻒ ﻣﺸﺘﻖﭘﺬﯾﺮي ﺗﻌﻤﯿﻢﯾﺎﻓﺘﮥ ﯾﮏ ﺗﺎﺑﻊ ﺣﻘﯿﻘﯽ ﯾﮏ ﻣﺘﻐﯿﺮي, و ﻣﺸﺘﻖ ﺗﻌﻤﯿﻢﯾﺎﻓﺘﮥ آن ﻣﯽﭘﺮدازﯾﻢ، ﺑﻪ ﮔﻮﻧﻪاي ﮐﻪ ﻣﺠﻤﻮﻋﮥ ﺗﺎﺑﻊﻫﺎي ﻣﺸﺘﻖﭘﺬﯾﺮ اﻓﺰاﯾﺶ ﻣﯽﯾﺎﺑﺪ و اﻋﺘﺒﺎر ﻗﻀﯿﻪﻫﺎي اﺳﺎﺳﯽ ﻧﻈﺮﯾﮥ ﺗﺎﺑﻊﻫﺎي ﻣﺸﺘﻖﭘﺬﯾﺮ ﯾﮏ ﻣﺘﻐﯿﺮي، ﻣﺎﻧﻨﺪ ﻗﻀﯿﮥ رول، ﻗﻀﯿﮥ ﻣﻘﺪار ﻣﯿﺎﻧﮕﯿﻦ ﮐﻮﺷﯽ، ﻗﻀﯿﮥ ﻣﻘﺪار ﻣﯿﺎﻧﮕﯿﻦ ﺑﺮاي ﻣﺸﺘﻖ، و ﻗﻀﯿﮥ ﺗﯿﻠﻮر ﺑﺮﻗﺮار ﺑﺎﻗﯽ ﻣﯽﻣﺎﻧﺪ. ﺳﺮاﻧﺠﺎم، ﻣﻘﺎﻟﻪ را ﺑﺎ اراﺋﮥ ﭼﻨﺪ ﻣﺜﺎل ﺑﻪ ﭘﺎﯾﺎن ﻣﯽﺑﺮﯾﻢ.

The current definition of the derivative makes the set of differentiable functions much smaller than the set of continuous functions, such that most of the real single variable functions are not differentiable and the surveying the rate of their growth is not possible with the available definition. In the present paper, using Caratheodory definition, we extend the set of differentiable functions by introducing a definition for generalized differentiation of a single variable function and its generalized derivative, in such a way that the validity of the basic theorems of this theory such as Rolle's theorem, Cauchy's mean value theorem, mean value theorem and Taylor's theorem would be hold. Finally we give some examples.