    f:R→R f(x+y) = f(x)+f(y) x,y   f

x+y
2

=
f(x)+f(y)
2
.   f:R→R x,y∈R   f  c f(2x) = 2f(x) x∈R x 1
2
x f

1
2
x

=
1
2
f(x).
  g(x) = f(x)+c g

x+y
2

= f

x+y
2

+c =
1
2

f(x)+f(y)

+
1
2
(2c) =
g(x)+g(y)
2
.
Obzornik mat. ﬁz. 55 (2008) 6
Peter Šemrl
 f:R → R  c
  x7→ f(x)+c  g(x) = f(x)−f(0) g

x+y
2

=
g(x)+g(y)
2
, x,y∈R. y = 0 g(0) = 0 g(x/2) =
(1/2)g(x) x∈R g(x+y) = g(x)+g(y) x,y∈R       f:R → R  f(x+y)≤ f(x)+f(y), x,y∈R.  f:R→R f

x+y
2

≤
f(x)+f(y)
2
, x,y∈R,  f f

tx+(1−t)y

≤ tf(x)+(1−t)f(y) x,y t ∈ [0,1]  t = 1/2     f:R→R f
2
  a b 2ab≤ a
2
+b
2
 
f

x+y
2

2
≤

f(x)+f(y)
2

2
=

f(x)

2
+2f(x)f(y)+

f(y)

2
4
≤
≤

f(x)

2
+

f(y)

2
2
.
 Obzornik mat. ﬁz. 55 (2008) 6
O dveh funkcijskih enaˇ cbah in ustreznih neenaˇ cbah
   f(x) = |x| x 7→ x
2
  f f(2x) ≤ 2f(x) f(3x) = f(x + 2x) ≤
3f(x)  f(nx) ≤ nf(x) n ∈ N x∈R  f [0,1]
  M x > 1 [x]  x x f(x) = f


[x]+1

x
[x]+1

≤

[x]+1

M ≤ (x+1)M < 2Mx.
 (1,∞)  f(x)/x   g(x) = e
x
  x
n
 x ∞  g x y g

1
4
x+
3
4
y

= g
 
x+y
2
+y
2
!
≤
g
 
x+y
2

+g(y)
2
≤
≤
g(x)+g(y)
2
+g(y)
2
=
1
4
g(x)+
3
4
g(y).
 g(tx+(1−t)y)≤ tg(x)+(1−t)g(y)
 t t =
k
2
n
,
 n k < 2
n
 [0,1] g   f(x) =

1; x≤ 0
e
−x
2
; x≥ 0
 x ≤ 0 y ≤ 0 f(x) = 1 f(y) = 1 0 < f(z) ≤ 1 z f(x+y)≤ f(x)+f(y) x,y > 0 f(x+y) = e
−(x+y)
2
< e
−x
2
= f(x) < f(x)+f(y).
209–212
Peter Šemrl
 J   f
′′
(x) = 2e
−x
2
(2x
2
−1) f
′′
 
0,
1
√
2

  f   f:R → R x
0
 δ M |f(x)|≤ M x∈ (x
0
−δ,x
0
+δ)  ℄  f(x) =

0; x∈Q
1; x6∈Q
     ℄ Obzornik mat. ﬁz. 55 (2008) 6