Continuity at a Point

Question

Show that f(x) is continuous at x=a if and only if lim(h→0) f(a+h) = f(a).

 

 

How?

 

 

Answer 1

By definition of continuity, a function f : R → R is continuous at a ∈ R, iff lim {x → a} f(x) = f(a)

Let x = a + h

Then, f is continuous at a iff lim {a + h → a} f(a + h) = f(a) . . . . . . substituting into the above definition

iff lim {h → 0} f(a + h) = f(a)

Done!