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

Examples 6.3.8(a):

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What kind of discontinuity does the function f(x) = exp(1/x) have at x = 0 ?

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f(x) = exp(1/x)

As x approaches zero from the right, 1/x approaches positive infinity. Therefore, the limit of f(x) as x approaches zero from the right is positive infinity.

As x approaches zero from the left, 1/x approaches negative infinity. Therefore, the limit of f(x) as x approaches zero from the left is zero.

Since the right-handed limit fails to exist, the function has an essential discontinuity at zero.