i was reading my notes and it stated this def for continuity:

f is continuous if for all ε>0 there exist δ>0 such that 0< l x-c l <δ, then lf(x) -f(c) l < ε.

i thought that by saying 0< l x-c l <δ, we are considering the cases where x cant be c. which is that the limit is not f(c)..

but from this lf(x) -f(c) l < ε, my professor is already assuming that the limit is f(c) right?

and so its sufficient in the definition to just say l x-c l <δ... isit true?