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
• Java Tools

Theorem 6.5.5: Differentiable and Continuity

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If f is differentiable at a point c, then f is continuous at that point c. The converse is not true.

Context

Proof:

Note that

• f(x) - f(c) = ( x - c )

As x approaches c, the limit of the quotient exists by assumption and is equal to f'(c), and the limit of the right-hand factor exists also and is zero. Therefore:

• f(x) - f(c) = 0

which is another way of stating that f is continuous at x = c.