Insertion of a Function : Insertion of a function belonging to a certain subclass

1-Insertion of a function belonging to a certain subclass of ${\bbb r}^x$}. 2-Insertion of a continuous function between two comparable precontinuous (semi-continuous) functions. 3-Strong insertion of a continuous function between two comparable precontinuous (semi-continuous )functions. 4-Insertion of a contra-continuous function. 5-Strong insertion of a contra-continuous function. 6-insertion of an $\alpha-$continuous function. 7-Insertion of a continuous function between two comparable $\alpha $continuous ($c-$continuous) functions.