MathDB
Problems
Contests
International Contests
KoMaL A Problems
KoMaL A Problems 2017/2018
A. 723
A. 723
Part of
KoMaL A Problems 2017/2018
Problems
(1)
KöMaL A. 723
Source: KöMaL A. 723
5/12/2018
Let
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
be a continuous function such that the limit
g
(
x
)
=
lim
h
→
0
f
(
x
+
h
)
−
2
f
(
x
)
+
f
(
x
−
h
)
h
2
g(x)=\lim_{h\rightarrow 0}{\frac{f(x+h)-2f(x)+f(x-h)}{h^2}}
g
(
x
)
=
h
→
0
lim
h
2
f
(
x
+
h
)
−
2
f
(
x
)
+
f
(
x
−
h
)
exists for all real
x
x
x
. Prove that
g
(
x
)
g(x)
g
(
x
)
is constant if and only if
f
(
x
)
f(x)
f
(
x
)
is a polynomial function whose degree is at most
2
2
2
.
real analysis
college contests