Abstract: We study the continuity of the definite bound function of continuous function on the variable interval. With correspondence between the variable interval and the unit interval, the definite bound function on the change interval is expressed as the definite bound function on the unit interval. With the subtraction inequality of upper bounds of two functions and the uniform continuity of the function, it is proved that the definite bound function is uniformly continuous.

Key words: variable interval, definite bound function, continuous function, uniformly continuous