I understand how the Intermediate Value Theorem proves a sign change can only happen at roots if a function is continuous. Intuitively, I understand the possible sign change due to a discontinuity. However, I don't understand how the theorem proves a sign change can happen at a discontinuity when the theorem only applies to continuous functions. Or is that the reason that the theorem only applies to continuous functions? Thanks!
How does the Intermediate Value Theorem prove change of sign at discontinuity?
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$\begingroup$
calculus
discontinuous-functions
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0Could you please explain what you mean by "a sign change can only happen at roots"? – 2017-02-12
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0A function can only change sign at roots, or x-intercepts. – 2017-02-12
1 Answers
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The negation of that statement is that, when $f(a) < 0 < f(b)$ and $f$ does not have a root (in $(a,b)$), then $f$ is not continuous on $[a,b]$, i.e. there exists at least one point in that interval where $f$ is not continuous.