If f ( x ) = 1 ( x − 2 ) ( 2 x − 5 ) and g ( x ) = 1 x 2 , we define f(g(x)) over the domain in which its defined , is it correct to say f(g(x)) is discontinuous at + − √ ( 2 / 5 ) ,0 and +- 1 / √ 2 ? 0 because the input is going into g(x) hence as its not in domain of g(x) so it can be said its discontinuous at that point . and similarily other four points as its not in domain of f(g(x)) ? But if suppose we say h(x)=f(g(x)) will we say h(x) is discontinuous at those points (maybe different ones) where x is not defined after simplyfing the whole f(g(x)) to a rational function ?

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If   f  (  x  )  =      1          (      x      −      2      )      (      2      x      −      5      )       and   g  (  x  )  =      1          x      2       , we define f(g(x)) over the domain in which its defined , is it correct to say f(g(x)) is discontinuous at   +  −      √    (  2      /    5  ) ,0 and +-  1      /        √    2 ? 0 because the input is going into g(x) hence as its not in domain of g(x) so it can be said its discontinuous at that point . and similarily other four points as its not in domain of f(g(x)) ? But if suppose we say h(x)=f(g(x)) will we say h(x) is discontinuous at those points (maybe different ones) where x is not defined after simplyfing the whole f(g(x)) to a rational function ?

tangouwn · Accepted Answer

The definition of &quot;h is continuous at a&quot; is      lim          x      →      a        h  (  x  )  =  h  (  a  )For this to be true, three things have to be satisfied:1. h(x) must converge to some limit L as   x  →  a.2. h(a) must be defined. That is, a is in the domain of h.3. The values h(a) and L must be the same.Since none of   ±            2      5        ,  ±      1          2        ,  0 are in the domain of h, h is not continuous at any of those points.But does that mean the same thing as saying &quot;h is discontinuous&quot; at those points? I would argue no. If we take &quot;h is discontinuous at a&quot; to be exactly the negation of &quot;h is continuous at a&quot;, then we can equally well say &quot;h is discontinuous at Sally&quot;, because Sally is also not in the domain of h.But that is silly. h is defined on a subset of the real numbers, so most things are not going to be in the domain of h. Saying &quot;h is discontinuous at a&quot; should mean something significant about the behavior of h, not be something trivial and useless. So to say &quot;h is discontinuous at a&quot; should mean h is defined at a, but either does not converge there, or else converges to some other value than h(a).That is my opinion. But I am not the Lord High Arbiter of Mathematical Language. There are doubtless others who disagree. Some things are not quite important enough for a general consensus to develop.

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