The problem of simultaneous Chebyshev approximation of a set
F
F
of uniformly bounded, real-valued functions on a compact interval
I
I
. by a set
P
P
of continuous functions is equivalent to the problem of simultaneous approximation of two real-valued functions
F
+
(
x
)
,
F
−
(
x
)
{F^ + }(x),{F^ - }(x)
, with
F
−
(
x
)
≦
F
+
(
x
)
{F^ - }(x) \leqq {F^ + }(x)
, for all
x
x
in
I
I
, where
F
−
{F^ - }
is lower semicontinuous and
F
−
{F^ - }
is upper semicontinuous.