what does the function involving rho mean in this definition?
While the pictures demonstrate the notions of continuity and discontinuity at a real value, with graph in the plane, there is nothing in this definition that is specific to real functions of one variable. The [tex] \rho [/tex] in the definition refers to the metric (distance function) defined for the range of the function [tex] f [/tex].
so rho(f(x),b) in this definition is the distance between f(x) and b in the range of f right?
Yes. For real-valued functions as usually studied in calculus, both [tex] d [/tex] and [tex] \rho [/tex] are the absolute value, but in other situations other distance functions can be used.
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