Question

1. Suppose that a function f, defined for all real numbers, satisfies the property that, for all x and y, f(x + y) = f(x) + f(y). (*) (a) (2 points) Name a function that satisfies property (*). Name another that doesn't. Justify your answers. (b) (3 points) Prove that any function that satisfies property (*) also satisfies f(3x) = 3f(x). (c) (5 points) Prove that any function that satisfies property (*) also satisfies f(x - y) = f(x) – f(y). (d) (5 points) Suppose in addition to (*) that f is differentiable at x = 0. Prove that f is differentiable everywhere.

Answer 1

@ Defined fw sa fol+ y)2 % + y2 f3+ fly) ft satisfy o then clearly Define then f2 (A) = x² fa does not satisfy fcd) + f1 (2) = 1 +2²2 5 f2 (3) = fz (2+1) het f satisfy . then fra fx+x)=" fra+ fruit flas f(3x) 3 fixes. to we know fosis = fcel-y)+4) fil-y) - fears - 'fly) fix-y) + fly, @ Given f is differentiable at azo lim flos-fios exists ² lim fix) exists. za 270 2-0 2 Now to check differentia blity differentia blity aut uza. 2-a xna yoo y lim fixe - flas clim f(x-a) nya lim fitra) - lim fiy letting y=x-a (1.2427o ana line fine exists & lim flyl exist & lim fini-foil exist yao f is differentiable at isa 2120 x 3679 7-9 f is differentiable and of everywhere