Given a continuous function f : R → R with: lim x → ∞ f ( x ) = 0 lim x → − ∞ f ( x ) = 0 f ( x ) &gt; 0 ∀ x ∈ R Show that there exists x , y ∈ R with x ≠ y so that f ( x ) = f ( y ).In this question no interval is given. What I want to do is to show that there is a maximum and then use the Intermediate value theorem. But I don't know how to do that without any interval given. Is it sufficient enough to say that because the limit for x to infinity and -infinity and f(x)&gt;0 there must exist a maximum value? Since f(x) cannot be zero.

Question

Given a continuous function   f  :      R    →      R   with:      lim          x      →      ∞        f  (  x  )  =  0      lim          x      →      −      ∞        f  (  x  )  =  0  f  (  x  )  &amp;gt;  0    ∀    x  ∈      R  Show that there exists   x  ,  y  ∈      R   with   x  ≠  y so that   f  (  x  )  =  f  (  y  ).In this question no interval is given. What I want to do is to show that there is a maximum and then use the Intermediate value theorem. But I don&#039;t know how to do that without any interval given. Is it sufficient enough to say that because the limit for x to infinity and -infinity and f(x)&amp;gt;0 there must exist a maximum value? Since f(x) cannot be zero.

Bornejecbo · Accepted Answer

Take any       x    0   to start. Then   f  (      x    0    )  &amp;gt;  0 and       lim          x      →      −      ∞        f  (  x  )  =      lim          x      →      ∞        f  (  x  )  =  0 imply that1. there is an   x  &amp;lt;      x    0   such that   f  (  x  )  =            f      (              x        0            )        2  , because as   x goes from       x    0   to   −  ∞ and   f  (  x  ) vanishes, the value of   f  (  x  ) must pass through   ϵ, for each   ϵ satisfying   f  (      x    0    )  &amp;gt;  ϵ  &amp;gt;  0.2. there is a   y  &amp;gt;      x    0   such that   f  (  y  )  =            f      (              x        0            )        2  .There is your   x  ,  y.

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