Exercise · ZW-CONTINUOUS-FUNCTION-019 · ★★★ Advanced
If \(f\) is continuous at \(c\) and \(f(c)>0\), what can be guaranteed?
Choose one option.
Show answer
- A. There is \(\delta>0\) with \(f(x)>0\) whenever \(|x-c|<\delta\)
Explanation
Taking \(\varepsilon=f(c)/2\) gives \(f(x)>f(c)/2>0\) near \(c\). Neither \(f(x)>f(c)\) nor anything outside a neighborhood follows; without continuity it fails (for example with a jump).
Review the concept: Continuous function
A function \(f\) is continuous at \(a\) if \(f(a)\) is defined, its limit as \(x\) tends to \(a\) exists and equals \(f(a)\).
\[\lim_{x\to a}f(x)=f(a)\]
More practice problems: continuous function
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