Sequential Definitions of Continuity for Real Functions

Ideally slowly oscillating sequences

“…Connor and Grosse-Erdman [20] gave sequential definitions of continuity for real functions calling G-continuity (see [10]) instead of A-continuity and their results covers the earlier works related to A-continuity where a method of sequential convergence, or briefly a method, is a linear function G defined on a linear subspace of s, space of all sequences, denoted by c G , into R. A sequence x = (x n ) is said to be G-convergent to ℓ if x ∈ c G and G(x) = ℓ. In particular, lim denotes the limit function lim x = lim n x n on the linear space c and st-lim denotes the statistical limit function st-lim x = st-lim n x n on the linear space st(R).…”

On arithmetic continuity

“…Using the idea of continuity of a real function and the idea of compactness in terms of sequences, we introduce the concept of arithmetic continuity and arithmetic compactness and establish some interesting results related to these notions. For details on continuity of real valued functions we refer to [1,4,5,6,10] 2. Arithmetic Continuity Definition 2.1.…”

Variations on strong lacunary quasi-Cauchy sequences

“…, where a sequence x = (x n ) is said to be Gconvergent to L if x ∈ c G , and G(x) = L ( [29]). A method G is called regular if every convergent sequence x is G-convergent with G(x) = lim x.…”