### ON INVARIANT CONTINUITY AND INVARIANT COMPACTNESS IN BANACH SPACES

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#### Abstract

In this study, we have defined the concepts of invariant continuity, invariant compactness, invariant boundedness and invariant Cauchy sequence in normed linear spaces. In general, there is no relation between continuity and invariant continuity. We have proved that if f is a linear map, then continuity of f implies invariant continuity of f. Additionally, we have shown that continuity of f and invariant continuity of f is equal under a condition. Also, we have proved that every invariant convergent sequence is invariant Cauchy. Finally, we have proved that invariant continuous image of an invariant compact space is invariant compact.

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DOI: https://doi.org/10.22190/FUMI230525010S

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