1
a=0 b=0대입
f(f(0))=f(0)
a=f(0) b=0대입
f(f(f(0)))=f(0)+f(0)
f(0)=2f(0) f(0)=0

b=0대입
f(f(a))+0)=a+f(0)
f(f(a))=a

2.
limf(x)=f(2020)
x→2020

f(f(a)+b)=a+f(b) a에 f(a)대입
f(f(f(a))+b)=f(a)+f(b)
f(a+b)=f(a)+f(b)

임의의 a에 대해
limf(x)=f(a)증명 x-a+2020=t
x→a

limf(t+a-2020)=f(a)보이면됨
t→2020

f(t+a-2020)=f(t)+f(a-2020)=f(t)+f(-2020)+f(a)

f(2020-2020)=f(2020)+f(-2020)
f(-2020)=f(0)-f(2020)=-f(2020)이므로

f(t+a-2020)=f(t)-f(2020)+f(a)

limf(t+a-2020)=limf(t)-f(2020)+f(a)=f(a)
t→2020             t→2020

3.
미분가능

lim{f(2020+h)-f(2020)}/h =f'(2020) 존재
h→0

동치
lim f(h)/h 
h→0

임의의 a에대해
lim{f(a+h)-f(a)}/h=f'(a) 존재함을 보이자
h→0

동치

limf(h)/h =f'(2020)
h→0

f'(x)=f'(2020)

f(x)=f'(2020)x

f(f(x))=f'(2020)f(x)=x

f(x)=x/f'(2020)

f'(2020)=1/f'(2020)

1) f'(2020)=-1 일 때
f(x)=-x
2) f'(2020)=1 일 때
f(x)=x





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