Let ` f(x+y) = f(x) + f(y)` for all x and y. If the function f (x) is continuous at x = 0 , then show that f (x) is continuous at all x.

Let ` f(x+y) = f(x) + f(y)` for all x and y. If the function f (x) is

1.	Let ` f(x+y) = f(x) + f(y)` for all x and y. If the function f (x) is continuous at x = 0 , then show that f (x) is continuous at all x.
Answer» Since, f(x) is continuous at x = 0. `rArr" " underset( x to 0) lim f(x) = f(0) ` `rArr" "f(0^(+)) = f(0^(-))= f (0) = 0` ….(i) To show, continuous at x = k RHL = ` underset( h to 0) lim f(k+h) = underset( h - 0) lim [f(k)+f(h)] = f(k) + f(0^(+))` ` " " f(k) + f(0) ` LHL = ` underset( h to 0) lim f(k - h) = underset( h to 0) lim [f(k) + f(-h)]` ` " " = f(k) + f(0^(-)) = f(k) + f(0) ` ` :." " underset( x to k) lim f(x) = f(k) ` ` rArr f(x) ` is continuous for all ` x in R`.