関数の話 - Speaker Deck

関数の話

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ؔ਺ͷ࿩ !ZVSPZPSP

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ؔ਺ͱ͸

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Z G Y ೖྗYʹରͯ͠ɺग़ྗZͷ஋Λܾఆ͢Δنଇ

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" # $ % *OU 4USJOH ࣸ૾ͭͷू߹ͷରԠؔ܎

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JOU TUS JOH ू߹ܕͱߟ͑ΔͱɺܕͱܕͷରԠنଇ

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ؔ਺͕ ϑΝʔετΫϥε

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MFUGaY TIPX Y ม਺ʹʮؔ਺ʯΛ୅ೖ ଋറ Ͱ͖Δ )BTLFMM

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G Y \YUP@T ^ ม਺ʹʮؔ਺ʯΛ୅ೖ ଋറ Ͱ͖Δ 3VCZ

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DPOTUG Y \SFUVSOYUP4USJOH ^ ม਺ʹʮؔ਺ʯΛ୅ೖ ଋറ Ͱ͖Δ +BWB4DSJQU

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ؔ਺ద༻

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ධՁʮؔ਺ʯʹҾ਺Λ༩͑ͯ݁ՌΛಘΔ͜ͱ SFUVSOYUP4USJOH GJOUTUSJOH SFUVSO ʠʡ Ҿ਺Y

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G H I 2Ͳͷؔ਺͔ΒʮධՁʯ͢Δ

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G H I ਖ਼֨ධՁ಺ଆ͔Βɻී௨ͷݴޠ $+43VCZ+BWBʜ

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G H I ඇਖ਼֨ධՁ֎ଆ͔Βɻ஗ԆධՁ )BTLFMM*0ʜ

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ߴ֊ؔ਺

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MFUH *OU4USJOH TUSJOH HBʠIFMMPʠ B ʮؔ਺ʯΛड͚औΔؔ਺ )BTLFMM

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H GO \ʠIFMMPʠGO ^ ʮؔ਺ʯΛड͚औΔؔ਺ 3VCZ

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DPOTUH GO \SFUVSOʠIFMMPʠGO ^ ʮؔ਺ʯΛड͚औΔؔ਺ +BWB4DSJQU

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ʮؔ਺HʯͷҾ਺GOʹʮؔ਺GʯΛ༩͑Δ SFUVSOYUP4USJOH SFUVSOʠIFMMPʠGO GJOUTUSJOH H JOUTUSJOH TUSJOH lIFMMPz SFUVSO Ҿ਺GO

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ड͚औͬͨʮؔ਺ʯGOΛ࢖ͬͯTUSJOHΛฦ͢ H JOUTUSJOH TUSJOH SFUVSOʠIFMMPʠ SFUVSOYUP4USJOH GOJOUTUSJOH lIFMMPz SFUVSO Ҿ਺Y SFUVSOYUP4USJOH GOJOUTUSJOH

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Ϋϩʔδϟ

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MFUB MFUIaYYB पғͷʮ؀ڥʯΛऔΓࠐΜͩʮؔ਺ʯ )BTLFMM

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B I Y \YB^ पғͷʮ؀ڥʯΛऔΓࠐΜͩʮؔ਺ʯ 3VCZ

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DPOTUB DPOTUI Y \SFUVSOYB^ पғͷʮ؀ڥʯΛऔΓࠐΜͩʮؔ਺ʯ +BWB4DSJQU

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ʮؔ਺Iʯ͸ʮBʯͱ͍͏؀ڥΛ͍֮͑ͯΔ SFUVSOY IJOUJOU B B ؀ڥΛऔΓࠐΉ

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ʮؔ਺ʯ͕ʮঢ়ଶʯΛ͍࣋ͬͯΔͱ΋ݴ͑Δ SFUVSOY IJOUJOU B SFUVSO

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Ϋϩʔδϟͱ ߴ֊ؔ਺

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MFUJ*OU *OU4USJOH JYaZTIPX YZ Ҿ਺ΛऔΓࠐΜͩʮΫϩʔδϟʯΛ࡞ͬͯฦ͢ʮؔ਺ʯ )BTLFMM Ϋϩʔδϟ

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J Y \ Z \ YZ UP@T^ ^ Ҿ਺ΛऔΓࠐΜͩʮΫϩʔδϟʯΛ࡞ͬͯฦ͢ʮؔ਺ʯ 3VCZ Ϋϩʔδϟ

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DPOTUJ Y \ SFUVSO Z \ SFUVSO YZ UP4USJOH ^ ^ Ҿ਺ΛऔΓࠐΜͩʮΫϩʔδϟʯΛ࡞ͬͯฦ͢ʮؔ਺ʯ +BWB4DSJQU Ϋϩʔδϟ

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ड͚औͬͨҾ਺ΛऔΓࠐΜͩʮΫϩʔδϟʯΛฦ͢ JJOUJOUTUSJOH SFUVSO JOUTUSJOH SFUVSO Y ؀ڥΛऔΓࠐΉ SFUVSO Z UP4USJOH Y Y SFUVSO Z UP4USJOH Ҿ਺Y JOUTUSJOH

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ؔ਺߹੒

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GɾH Y G H Y ̎ͭͷؔ਺G HΛॱ൪ʹద༻͢Δ߹੒ؔ਺GɾHΛੜ੒͢Δ͜ͱ

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MFUGaYY MFUHaYTIPXY MFUHGaYHGY ؔ਺HG͸Ҿ਺YΛGʹద༻ͨ݁͠ՌΛHʹద༻ͯ͠ฦ͢ )BTLFMM

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G Y \Y^ H Y \YUP@T^ HG Y \H G Y ^ ؔ਺HG͸Ҿ਺YΛGʹద༻ͨ݁͠ՌΛHʹద༻ͯ͠ฦ͢ 3VCZ

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DPOTUG Y \SFUVSOY^ DPOTUH Y \SFUVSOYUP4USJOH ^ DPOTUHG Y \SFUVSOH G Y ^ ؔ਺HG͸Ҿ਺YΛGʹద༻ͨ݁͠ՌΛHʹద༻ͯ͠ฦ͢ +BWB4DSJQU

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߹੒ؔ਺HG͸ؔ਺GͱHΛ΋ͭʮΫϩʔδϟʯ SFUVSO HGJOUTUSJOH SFUVSOY GJOUJOU SFUVSOYUP4USJOH HJOUTUSJOH SFUVSOY GJOUJOU SFUVSOYUP4USJOH HJOUTUSJOH

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߹੒ؔ਺HG͸ؔ਺GͱHΛ΋ͭʮΫϩʔδϟʯ SFUVSO HGJOUTUSJOH SFUVSOY GJOUJOU SFUVSOYUP4USJOH HJOUTUSJOH SFUVSO ʠʡ

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ؔ਺߹੒͢Δؔ਺

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DPNQPTF G H GɾH ̎ͭͷؔ਺G HΛҾ਺ʹ߹੒ؔ਺HGΛฦ͢ʮؔ਺ʯ

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CD BC BD MFUHGHG )BTLFMMʹ͸૊ΈࠐΈͰ ؔ਺͕͋Δ )BTLFMM

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EFGDPNQPTF H G Y \H G Y ^ FOE 3VCZͰ͸Ҿ਺H G͔ΒMBNCEBΛੜ੒ͯ͠ฦ͢ 3VCZ

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DMBTT1SPD EFGDPNQPTF G Y \TFMG G Y ^ FOE BMJBT@NFUIPE DPNQPTF FOE 1SPDΫϥεͷΠϯελϯεϝιουʹ͢Δͷ͕ࣗવ 3VCZ

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G Y \Y^ H Y \YUP@T^ HDPNQPTF G HG 1SPDΫϥεͷΠϯελϯεϝιουʹ͢Δͷ͕ࣗવ 3VCZ

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DPOTUDPNQPTF H G \ SFUVSO Y \SFUVSOH G Y ^ ^ +BWB4DSJQUͰ͸Ҿ਺H G͔ΒGVODUJPOΛੜ੒ͯ͠ฦ͢ +BWB4DSJQU

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'VODUJPOQSPUPUZQFDPNQPTF GVODUJPO G \ SFUVSO Y \SFUVSOUIJT G Y ^ ^ 'VODUJPOʹDPNQPTFΛఆٛ͢Δํ͕ࣗવ +BWB4DSJQU

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ؔ਺ W T ΦϒδΣΫτ

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Φ ϒδΣ Ϋ τ ؔ ਺ ม ਺ ʹ୅ ೖͰ͖ Δ ม ਺ ʹ୅ ೖͰ͖ Δ

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Φ ϒδΣ Ϋ τ ؔ ਺ ؔ ਺ ͷҾ ਺ʹ౉ ͤ Δ ؔ ਺ ͷҾ ਺ʹ౉ ͤ Δ

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Φ ϒδΣ Ϋ τ ؔ ਺ ঢ় ଶ Λ࣋ ͭ ϝϯό ʔ ม਺ ঢ় ଶ Λ࣋ ͭ Ϋϩʔ δ ϟ

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Φ ϒδΣ Ϋ τ ؔ ਺ ؔ ਺ Ͱੜ ੒ͯ͠ ໭ Γ ஋ʹ Ͱ͖Δ ؔ ਺ Ͱੜ ੒ͯ͠ ໭ Γ ஋ʹ Ͱ͖Δ

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Φ ϒδΣ Ϋ τ ؔ ਺ ૊ Έ ߹Θ ͤΔ͜ ͱ ͕Ͱ͖ Δ EF M FH BU F D PN Q PTJ UJP O ૊ Έ ߹Θ ͤΔ͜ ͱ ͕Ͱ͖ Δ ؔ ਺߹੒

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ؔ਺͜Θ͘ͳ͍ ͜Θ͘ͳ͍

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͔͜͜Β͸ ͕࣌ؒ༨ͬͨΒ

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ؔ਺߹੒ͷ ॱং

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I H G Y YΛGʹ౉ͨ݁͠ՌΛɺHʹ౉ͯ͠ಘͨ݁ՌΛɺIʹ౉͢

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I H G Y ӈ͔Βࠨ΁ॲཧ͕ྲྀΕ͍ͯ͘

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IɾHɾG Y ߹੒ؔ਺IɾHɾG͸ӈ͔Βద༻͞Ε͍ͯ͘

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GH Y H G Y ߹੒ͷॱংΛٯʹͯ͠߹੒͢Δ Λߟ͑Δ GɾH Y G H Y

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Ͱ߹੒͢Δͱɺॲཧ͕ࠨ͔ΒӈʹྲྀΕΔ I H G Y GHI Y

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JNQPSU$POUSPM$BUFHPSZ $BUFHPSZDBUDBUBCDBUCDDBUBD )BTLFMMͰ͸$POUSPM$BUFHPSZʹ ͕͋Δ )BTLFMM

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DMBTT1SPD EFG H Y \H TFMG Y ^ FOE FOE 3VCZͰ͸1SPDͷԋࢉࢠ Λॻ͖׵͑Δ 3VCZ

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'VODUJPOQSPUPUZQFBOE5IFO GVODUJPO G \ SFUVSO Y \SFUVSOG UIJT Y ^ ^ 'VODUJPOʹBOE5IFOͳͲ࡞Ε͹Ͳ͏ʹ͔ +BWB4DSJQU

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ؔ਺߹੒ͷॱংΛม͑Δͱ ࠨ͔Βӈ΁ॲཧͷྲྀΕΛ දݱ͢Δ͜ͱ͕Ͱ͖Δ ೔ຊਓͳͷͰࠨ͔ΒಡΈ͍ͨ

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ؔ਺͜Θ͘ͳ͍ ͜Θ͘ͳ͍

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ؔ਺ద༻͢Δ ؔ਺

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ؔ਺ద༻΋ࠨ͔Βӈʹྲྀ͍ͨ͠ GHI Y

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] ͸ؔ਺ద༻͢Δʮؔ਺ʯ Y]GG Y

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Y͕ࠨ͔ΒӈʹG H IΛ௨ͬͯྲྀΕ͍ͯ͘ Y] GHI

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] B BC C B]GGB தஔԋࢉࢠ ] ͸ӈลͷؔ਺ʹࠨลͷ஋Λ౉͢ )BTLFMM

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DMBTT0CKFDU EFGʛʼ G G TFMG FOE FOE 3VCZͰ͸શ֯ ʛʼ ͰͲ͏ʹ͔ʜ 3VCZ

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0CKFDUQSPUPUZQFSFWBQQMZ GVODUJPO G \SFUVSOG UIJT ^ 0CKFDUQSPUPUZQFԚછ +BWB4DSJQU

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ؔ਺ద༻͢Δؔ਺Ͱ Ҿ਺͕ࠨ͔Βӈ΁ྲྀΕΔ༷Λ දݱ͢Δ͜ͱ͕Ͱ͖Δ ೔ຊਓͳͷͰࠨ͔ΒಡΈ͍ͨ

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6OJYͷQJQF΍ &MJYJSͷ]ͳͲ Ͱ΋͋Μ·Γ͜͏͍͏ॻ͖ํ͠ͳ͍͚Ͳʜ

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ؔ਺͜Θ͘ͳ͍ ͜Θ͘ͳ͍

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ؔ਺߹੒ͷ จ຺

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EFGDPNQPTF G Y \TFMG G Y ^ FOE ؔ਺߹੒ͷ࣮૷Ͱʜ 3VCZ

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EFG G Y \TFMG G Y ^ FOE G Y ͷ݁ՌΛഒʹͯ͠ΈΔ 3VCZ

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G Y \Y^ H Y \Y^ I Y \YUP@T^ IHG ʠʡ ߹੒͞Εͨؔ਺ͷؒͰҾ਺͕ഒʹ 3VCZ

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߹੒͞Εͨؔ਺ͷؒͰҾ਺͕ഒʹ Y GJOUJOU Y HJOUJOU YUP@T IJOUTUSJOH ʠʡ

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EFG G Y \ SFTG Y SFTOJM OJMTFMG SFT ^ FOE G Y ͕OJMͳΒ͹ɺ߹੒͞Εͨؔ਺Λݺ͹ͣʹऴྃ͢Δ 3VCZ

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G Y \TBNQMF^ H Y \TBNQMF^ I Y \YUP@T^ IHG OJMPSʠʡ G HͷͲͪΒ͔͕OJMΛฦ͢ͱશମͷ݁Ռ͕OJMʹͳΔ 3VCZ

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߹੒͞Εͨؔ਺ͷؒͰOJMνΣοΫ TBNQMF GJOUJOU TBNQMF HJOUJOU YUP@T IJOUTUSJOH OJM OJM ʠʡ ऴྃ ऴྃ

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ʮ݁Ռ͕ഒʹͳΔʯจ຺ ʮOJMͩͱதࢭ͢Δʯจ຺ ؔ਺߹੒ʹ༷ʑͳʮจ຺ʯΛ༩͑Δ͜ͱ͕Ͱ͖Δ

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ʮঢ়ଶ͕มԽ͢Δʯจ຺ ʮೖग़ྗ͕ൃੜ͢Δʯจ຺ ʮඇಉظͰܭࢉ͢Δʯจ຺ ଞʹ΋͍Ζ͍Ζͳʮจ຺ʯ͕͋Δ

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ؔ਺߹੒Λந৅Խ͢Δͱ ੜ੒͞ΕΔؔ਺ʹ༷ʑͳ ੑ࣭Λ༩͑Δ͜ͱ͕Ͱ͖Δ ΋͔ͯ͠͠Ϟφʜʜ ҧ͍·͢

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ؔ਺͜Θ͘ͳ͍ ͜Θ͘ͳ͍

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ྫ ؔ਺߹੒Ͱ 'J[[#V[[

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DMBTT1SPD EFGPS G Y \ SFTTFMG Y SFTOJM G Y SFT ^ FOE FOE ࣗ਎ʹYΛద༻ͨ݁͠Ռ͕OJMͳΒ͹ɺ߹੒͞Εͨؔ਺ΛݺͿ 3VCZ

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GJ[[ Y \Y GJ[[OJM^ CV[[ Y \Y CV[[OJM^ GJ[[CV[[ Y \ YY GJ[[CV[[OJM^ JE Y \YUP@T^ GJ[[ CV[[ GJ[[CV[[ͱ਺ࣈΛग़ྗ͢Δؔ਺Λ༻ҙ 3VCZ

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NBQ GJ[[CV[[PSGJ[[PSCV[[PSJE ֤ؔ਺ΛPSͰ߹੒ͯ͠NBQ͢Ε͹GJ[[CV[[Ͱ͖Δ 3VCZ

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ͦͷଞͷ τϐοΫ

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wŋࢀরಁա wŋΧϦʔԽ wŋ෦෼ద༻ wŋ࠶ؼ wŋ৞ΈࠐΈؔ਺ wŋ୅਺σʔλߏ଄ ϞϊΠυͱ͔ ͜ͷ͋ͨΓͷ࿩΋ػձ͕͋ͬͨΒ

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ؔ਺͜Θ͘ͳ͍ ͜Θ͘ͳ͍