21f = x
g = 1/x
a = 0
WE hav discussed this once but a recent que. in Reso Test Series has forced me to ask agauin
true / false
if h(x) = f(x).g(x);
f(x) is cont. at x = a while g(x) is discont at x =a
then can h(x) be cont. at x = a ???
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14 Answers
21ya, so wen did i say i disagree wid u ..........
there mite b others who hav objection.........
21coz as far as i rem. last time we discussed that it h(x) CANT BE CONT. at a
Expert Confirmation needed then v can seal this un.
1isnt it like h(x)=f(x)g(x)
lim h(a+k)=f(a+k). g(a+k)
k→0
lim h(a-k)=f(a-k). g(a-k)
k→0
if f(x) is cont at a then lim f(a+k)=lim f(a-k)=f(a)
k→0 k→0
which is not applicable to g(x)
therefore h(a)=f(a).g(a)≠lim f(a+k).g(a+k)
k→0
or h(a)≠lim f(a-k).g(a-k)
k→0
therefore h(x) is discontinuious at x=a
i think that for f(x)=x.... and g(x)=1/x
h(x) is defined only in the domain common to both f and g
so x=0 is not in the domain of h(x) and hence has a discontinuity
66mrnobody1 is correct. The product of a continuous and a discontinuous function is necessarily discontinuous. In the example f(x)=x and g(x)=1/x, at x=0, the product has still a discontinuity, since the product is defined only for the common domain. Hence, the product f(x)g(x) is discontinuous at x=0.
However, the product of two discontinuous function can be continuous.
21Sir, can u giv an eg. of this case : the product of two discontinuous function can be continuous.
341I think the discussion has to take into account that there are discontinuities of two kinds - removable and non-removable (is that the right term?).
f(x) = 0 would render the function continuous if g(x) had a removable discontinuity.
21Sir, doess "removable discontinuity." imply functions wer a limit is defind lyk sinx / x ??
341Strictly no, because the function is not defined at that point. The function has to be defined at that point to classify it as removable.