1. Suppose that f : R → R is a function such that f (x + y) = f (x) + f ( y) for all x, y ∈ R. Prove that f has a limit at 0 iff f has a limit at every point c in . 2. Let f be a function defined on a...

1. Suppose that f : R → R is a function such that f (x + y) = f (x) + f ( y) for all x, y ∈ R. Prove that f has a limit at 0 iff f has a limit at every point c in
.

2. Let f be a function defined on a deleted neighborhood of a point c. Prove that lim_{x → c}
f (x) = L iff lim_{x → c
+}
f (x) = L and lim_{x → c –}
f (x) = L

3. Suppose functions f and g are defined on a deleted neighborhood of c. Prove that if lim<sub>x

à

c</sub>
f(x) =L >0 iff lim_x
à
c
g(x) =
, then lim_x
à
c
(fg)(x)=