Sentence examples for a counterexample is given from inspiring English sources

A counterexample is given as follows.

A counterexample is given to show that the latter result is not true in general.

In the following, a counterexample is given to show that the result in Theorem 3 is false.

However, if the gradient ∇f is assumed to be only Lipschitz continuous, then the sequence generated by the GPA converges weakly if H is infinite-dimensional (a counterexample is given in [1]).

If the gradient ∇f is only assumed to be Lipschitz continuous, then { x n } can only be weakly convergent if H is infinite-dimensional (a counterexample is given in Section 5 of Xu [33]).

However, there was a singular case; for this singularity, a counterexample was given by Gǎvruta [11].

A simple counterexample is given by: :(e^{2\pi i})^i=1^i=1\neq e^{-2\pi}=e^{2\pi\cdotot i} The identity is, however, valid when z is a real number, and also when u is an integer.

The counterexample is given by [2], from which one can see clearly that [1, Lemma  2.5] is wrong.

The counterexample was given by Yang [5].

See, for example, Yuan et al. [13], where various of counterexamples are given.

However, on the other hand, the results on E-convex programming in Youness [1] were not correct, and some counterexamples were given by Yang [5].