Real analysis definition question

[royzizzle]

what does the function involving rho mean in this definition?
(in picture)

 

[statdad]

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 \rho in the definition refers to the metric (distance function) defined for the range of the function f.

 

[royzizzle]

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 \rho in the definition refers to the metric (distance function) defined for the range of the function f.

so rho(f(x),b) in this definition is the distance between f(x) and b in the range of f right?

k

thanks!

 

[statdad]

so rho(f(x),b) in this definition is the distance between f(x) and b in the range of f right?

k

thanks!

Yes. For real-valued functions as usually studied in calculus, both d and \rho are the absolute value, but in other situations other distance functions can be used.