:: Real Function Differentiability
:: by Konrad Raczkowski and Pawe{\l} Sadowski
::
:: Received June 18, 1990
:: Copyright (c) 1990 Association of Mizar Users
theorem Th1: :: FDIFF_1:1
:: deftheorem Def1 defines convergent_to_0 FDIFF_1:def 1 :
reconsider cs = NAT --> 0 as Real_Sequence by FUNCOP_1:57;
:: deftheorem FDIFF_1:def 2 :
canceled;
:: deftheorem Def3 defines REST-like FDIFF_1:def 3 :
reconsider cf = REAL --> 0 as Function of REAL , REAL by FUNCOP_1:57;
:: deftheorem Def4 defines linear FDIFF_1:def 4 :
theorem :: FDIFF_1:2
canceled;
theorem :: FDIFF_1:3
canceled;
theorem :: FDIFF_1:4
canceled;
theorem :: FDIFF_1:5
canceled;
theorem Th6: :: FDIFF_1:6
theorem Th7: :: FDIFF_1:7
theorem Th8: :: FDIFF_1:8
theorem Th9: :: FDIFF_1:9
theorem Th10: :: FDIFF_1:10
theorem Th11: :: FDIFF_1:11
:: deftheorem Def5 defines is_differentiable_in FDIFF_1:def 5 :
:: deftheorem Def6 defines diff FDIFF_1:def 6 :
:: deftheorem Def7 defines is_differentiable_on FDIFF_1:def 7 :
theorem :: FDIFF_1:12
canceled;
theorem :: FDIFF_1:13
canceled;
theorem :: FDIFF_1:14
canceled;
theorem Th15: :: FDIFF_1:15
theorem Th16: :: FDIFF_1:16
theorem :: FDIFF_1:17
:: deftheorem Def8 defines `| FDIFF_1:def 8 :
theorem :: FDIFF_1:18
canceled;
theorem :: FDIFF_1:19
theorem Th20: :: FDIFF_1:20
theorem Th21: :: FDIFF_1:21
theorem Th22: :: FDIFF_1:22
theorem Th23: :: FDIFF_1:23
theorem Th24: :: FDIFF_1:24
theorem Th25: :: FDIFF_1:25
theorem :: FDIFF_1:26
theorem :: FDIFF_1:27
theorem :: FDIFF_1:28
theorem :: FDIFF_1:29
theorem :: FDIFF_1:30
theorem :: FDIFF_1:31
theorem Th32: :: FDIFF_1:32
theorem :: FDIFF_1:33
theorem Th34: :: FDIFF_1:34
theorem :: FDIFF_1:35
:: deftheorem DifDef defines differentiable FDIFF_1:def 9 :
Th3:
{} REAL is closed
Th4:
[#] REAL is open
theorem :: FDIFF_1:36