The following claim seems true for me:For any continuous function over reals f ( x → ), if f ( x → ) = c has no zero, then either f ( x → ) &gt; c for all x → or f ( x → ) &lt; c for all x → .However, my calculus is a bit rusty, and I am wondering from which this claim follows? (For univariate functions, this follows from the intermediate value theorem?)

Question

The following claim seems true for me:For any continuous function over reals   f  (            x      →        ), if   f  (            x      →        )  =  c has no zero, then either   f  (            x      →        )  &amp;gt;  c for all             x      →       or   f  (            x      →        )  &amp;lt;  c for all             x      →       .However, my calculus is a bit rusty, and I am wondering from which this claim follows? (For univariate functions, this follows from the intermediate value theorem?)

Cristopher Barrera · Accepted Answer

The standard proof of the one variable mean-value theorem works here as well. The relevant idea is that if   f is continuous, then   f maps connected sets to connected sets.So suppose you have   x and y such that   f  (  x  )  &amp;lt;  0 and   f  (  y  )  &amp;gt;  0. Then take a connected open set   U containing both   x and   y. As   f is continuous, we know   f  (  U  ) is connected and contains both a positive and negative number, and thus also contains 0.Thus your multivariate intermediate value theorem is true.

The following claim seems true for me: For any continuous function over reals f ( <mr

Answered question

Answer & Explanation

New Questions in Calculus 2