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A function with a derivative defined for all x, but whose derivative is discontinuous.

This gets a little involved. Define f by


For tex2html_wrap_inline205 we can find f'(x) by using the standard rules: it's tex2html_wrap_inline229 . This doesn't have a limit as x goes to 0: if you look at the sequence tex2html_wrap_inline233 (which goes to 0), tex2html_wrap_inline235 , but looking at the sequence tex2html_wrap_inline237 , tex2html_wrap_inline239 . Therefore tex2html_wrap_inline241 does not exist.

However, f'(0) does exist. Use the definition of derivative:


Since tex2html_wrap_inline247 , the difference quotient satisfies


for all tex2html_wrap_inline251 . But tex2html_wrap_inline253 , so


by the pinching theorem. Therefore f'(0) exists and equals 0.

Here's a picture of f:


Tom Vogel
Mon May 5 12:53:33 CDT 1997