a)Let f(x+y)=f(x)+f(y) for all x and y and if the function is continuous at x=0, then show that the function is continuous at all x.
b)If f(x.y)=f(x).f(y) for all x and the function is continuous at x=1. Prove that the function is ontinuous at all x except x=0. given f(1)not=0!
GIVE THE NECESSARY STEPS AS PER THE CONSIDERATION IF THE JEE WAS GOING TO BE SUBJECTIVE!!!!!!!!!!!!!!!!!!!!!
CHEERS!
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2 Answers
62f(x+y)=f(x)+f(y)
substitute y=0
so f(x)=f(x)+f(0)
so f(0)=0
for continuity,
lim(h->0) f(x+h)=f(x)
since, continuous as 0,
lim(h->0) f(0+h)=lim(h->0) {f(0)+f(h)} = f(0) + lim(h->0) (f(h))
since it is continuous at 0 lim(h->0) f(0+h)= f(0)
so f(0)=f(0)+lim(h->0) f(h)
thus, lim(h->0) f(h)=0
now we try continuity at some non zero point "k"
to prove
lim(h->0) f(k+h)=f(k)
lim(h->0) f(k+h)= lim(h->0) {f(k)+f(h)}
= lim(h->0)f(k)+lim(h->0)f(h)
=f(k)+0
thus it proves continuity at all k!
62for the second part...
hint:
f(x+h) tends to f(x)
f(x{1+h/x} = f(x) f(1+h/x)
this is the first step.. try using the method in the first part to solve this question!
