Continuity - Summary

To summarize, given a function y = f(x) defined at x = c, all the statements below are equivalent.

(1)    f is continuous at c.

(2)    Whenever x ≈ c, f(x) ≈ f(c).

(3)    Whenever st(x) = c, st(f(x)) = f(c).

(4)    lim_x→c f(x) = f(c).

(5)    y is continuous at x = c.

(6)    Whenever Δx is infinitesimal, Δy is infinitesimal.