Interactive Real Analysis - part of MathCS.org

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 6.1. Limits
• 6.2. Continuous Functions
• 6.3. Discontinuous Functions
• 6.4. Topology and Continuity
• 6.5. Differentiable Functions
• 6.6. A Function Primer
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
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Corollary 6.3.7: Discontinuities of Second Kind

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If f has a discontinuity of the second kind at x = c, then f must change from increasing to decreasing in every neighborhood of c.

Context

Proof:

Suppose not, i.e. f has a discontinuity of the second kind at a point x = c, and there does exist some (small) neighborhood of c where f, say, is always decreasing. But then f is a monotone function, and hence, by the previous theorem, can only have discontinuities of the first kind. Since that contradicts our assumption, we have proved the corollary.