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 6.5.6(c):

Why these ads ...

The function f(x) = x² sin(1 / x ) has a removable discontinuity at x = 0. If the function is extended appropriately to be continuous at x = 0, is it then differentiable at x = 0 ?

Back

• (f(x) - f(0)) / (x - 0) = x sin( 1 / x )

Since | x sin( 1 / x ) | < | x | for all x, we see that the limit of the difference quotient for c = 0 equals zero. Hence, f is differentiable at 0, and

• f'(0) = 0

The function is also differentiable everywhere else, since it is the product and composition of differentiable functions everywhere but for x = 0. Therefore, the function is differentiable on the whole real line.

Actually, this function is more interesting than it seems, because it is

• continuous
• once differentiable
• derivative is not continuous

Thus, it provides an example to show that even if derivatives exist, they do not necessarily have to be continuous. These statements are proved at a later point, but you might want to try it on your own already.