Question

Suppose that f satisfies f(x+y)=f(x)+f(y), and that f is continuous at 0. Prove that f is continuous on R

Answer 1

f(x) = kx (k=constant, k not equal to 0 )
f'(x)=k (which is never equal to 0)
which means f(x) will never have any maxima or minima --> continuous function
there are other methods to prove as well..

Answer 2

y=0,
f(x) = f(x + 0) = f(x) + f(0) ⇒ f(0) = 0,
therefore
0 = f(x - x) = f(x) + f(-x) ⇒ f(-x) = -f(x).

Let ε > 0 and x₀ ∈ R.
By continuity of f at x = 0 there exists δ > 0 such that
|x| < δ implies that |f(x)| < ε.
]For any x with |x - x₀| < δ,
|f(x) - f(x₀)| = |f(x - x₀)| < ε.

f is continuous.