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.