Continuous Functions that Fail to Be Differentiable
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We have seen that if is differentiable at , then is continuous at . Consequently, if is not continuous at then is differentiable at . For example :
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if and if
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Since is not continuous at , is not differentiable at .
However, a function can be continuous at a point where its derivative does not exists. Here we describe the ways in which a continuous function can fail to be differentiable.
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Recall that is differentiable at , if exists.
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Case (1) : or .
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For example: .
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f is continuous at 0, and
= .
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Hence, is not differentiable at . Nevertheless, the graph of f has a tangent line at (0, 0) --- a vertical tangent line.
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The graph of f near (0, 0) resembles a vertical line --- .
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Case (2) : ( ), ( ).
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For example : .
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f is continuous at 0, and
=
= .
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Hence, is not differentiable at and its graph has a sharp "cusp" at (0, 0). Nevertheless, the graph of f has a tangent line at (0, 0) --- a vertical tangent line.
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Here is the graph of f near (0, 0).
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Case (3): Both and exist, but
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.
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For example : .
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f is continuous at 0, and
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.
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is not differentiable at and its graph has a "corner point" at (0, 0) rather than a tangent line.
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Here is the graph of f near (0, 0).
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Case (4) : Either or does not exist.
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For example : , for all , and .
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f is continuous at 0, and does not exist.
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Here is the graph of f near (0, 0).
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In fact, the slopes of the secant lines oscillate between 1 and infinitely often and so the graph of f does not have a tangent line at (0, 0).
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Here some problems for you to think about :
1. Is there a function that is continuous everywhere and has "corner point" at each integer n , but is differentiable at every other point of the real line ?
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2. Is there a function that is continuous everywhere and has a vertical tangent line at each integer n , but is differentiable at every other point of the real line ?
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3. Suppose that the function f is continuous everywhere. At how many points do you suspect that f can fail to be differentiable ? What is the worst function you can think of ?
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