strong.
Secondly, if f (x
0
) > y
i+1
2
, let g ∈ G
←→
f
with the
constraint that g(x) = y
i+1
2
on JxK
I
S
. Such a function
g exists, because f
←→
≤ y
i+1
2
<
←→
f holds on JxK
I
S
.
Then, g
←→
= y
i+1
2
<
←→
f on JxK
I
S
, i.e., f
IFS
bw
6= g
IFS
bw
,
which contradicts the condition that f
IFS
bw
is roughly
strong.
Corollary 2. A rough real function f ∈ J
I
is (S
I
,P
J
)–
continuous on I if and only if Π
−
f
= Π
←→
f
.
Proof. f is roughly continuous
⇔ f
IFS
bw
is roughly strong by Proposition 8
⇔ f
IFS
pw
= f
IFS
bw
by Proposition 5
⇔ Π
−
f
= Π
←→
f
by Propositions 3 2.
Corollary 3. A rough real function f ∈ J
I
is (S
I
,P
J
)–
continuous on I if and only if G
−
f
= G
←→
f
.
Proof. f is roughly continuous
⇔ f
IFS
bw
is roughly strong by Proposition 8
⇔ f
IFS
pw
= f
IFS
bw
by Proposition 5
⇔ G
−
f
= G
←→
f
by Corollary 1 2.
5 CONCLUSION AND FUTURE
WORK
Rough continuity is a central notion in rough calculus.
This paper has characterized the rough continuity in
three different ways in terms of intuitionistic fuzzy
set theory.
This characterization establishes a connection be-
tween the two theories of uncertainty management,
the rough set theory and intuitionistic fuzzy set
theory. It may allow the application of the means of
intuitionistic fuzzy calculus in rough calculus.
In the future, the investigations can be continued
in several directions. This article has addressed only
one important concept of rough calculus, namely, the
rough continuity. First of all, rough continuity has
some additional features, such as rough discontinuity,
rough Darboux property or Intermediate Value
Property (IVP). The question is how they could also
be captured with the help of IFS tools. Moreover,
the relationships between additional notions of rough
calculus and IFS can also be studied.
Classical Pawlak’s rough set theory has many
different generalizations. The question is whether
they can be captured with IFS tools in one way or
another.
ACKNOWLEDGEMENTS
The author would like to thank the anonymous
referees for their useful comments and suggestions.
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