Interactive Real Analysis

• 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

6.3. Discontinuous Functions

Examples 6.3.8(b):

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

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f(x) = x sin(1/x)

Since | x sin(1/x) | < | x |, we can see that the limit of f(x) as x approaches zero from either side is zero. Hence, the function has a removable discontinuity at zero. If we set f(0) = 0 then f(x) is continuous.