A remark on discontinuity at fixed point

Contractive definitions and discontinuity at fixed point

“…Theorems 2.1, 2.2, and 2.3 unify and improve the results due to Bisht and Pant [1],Ćirić [5,6], Jachymski [8], Kuczma et al [10], Matkowski [11], and Pant [13].…”

“…Recently, Bisht and Pant [1] also gave a contractive definition which does not force the map to be continuity at the fixed point.…”

Proinov contractions and discontinuity at fixed point

“…Remark 1. The above proved theorems unify and improve the results due to Bisht and Pant [2], Boyd and Wong [4],Ćirić [7], Jachymski [8], Kannan [10], Lim [13], Kuczma et al [11], Matkowski [14], Pant [16,17], Park and Bae [19], Park and Rhoades [20], Proinov [21] and Rao and Rao [22].…”

Weak theta-phi-contraction and discontinuity

“…Let (X, d) be a metric space and T : X → X be a weak θ-φ-contraction, i.e, there exist θ ∈ Θ and φ ∈ Φ such that for any x, y ∈ X, T 2 x = T 2 y, 1) where…”

“…As pointed out in [1], in 1988, Rhoades [10] examined in detail the continuity of a large number of contractive mappings at their fixed points and demonstrated that though these contractive definitions do not require the map to be continuous yet the contractive definitions are strong enough to force the map to be continuous at the fixed point. So an interesting open question was raised by Rhoades [10] whether there is a contractive definition which is strong enough to generate a fixed point, but does not force the mapping to be continuous at the fixed point.…”