Continuous Functions that Fail to Be Differentiable

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We have seen that if f is differentiable at x[0] , then f is continuous at x[0] . Consequently, if f is not continuous at x[0] then f is differentiable at x[0] . For example :

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f(x) = x if x < 1 and f(x) = x+1 if 1 <= x

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Since f is not continuous at 0 , f is not differentiable at 0 .

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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 f is differentiable at x[0] , if limit((f(x[0]+h)-f(x[0]))/h,h = 0) exists.

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Case (1) : limit((f(x[0]+h)-f(x[0]))/h,h = 0) = infinity or -infinity .

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For example: f(x) = x^(1/3) .

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f is continuous at 0, and

limit((x^(1/3)-0)/(x-0),x = 0) = limit(1/(x^(2/3)),... = infinity .

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Hence, f is not differentiable at 0 . 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 --- x = 0 .

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Case (2) : limit((f(x[0]+h)-f(x[0]))/h,h = 0,left) = infinity ( -infinity ), limit((f(x[0]+h)-f(x[0]))/h,h = 0,right) = -infinit... ( infinity ).

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For example : f(x) = x^(2/5) .

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f is continuous at 0, and

limit((x^(2/5)-0)/(x-0),x = 0,left) = limit(1/(x^(3... = -infinity

limit((x^(2/5)-0)/(x-0),x = 0,right) = limit(1/(x^(... = infinity .

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Hence, f is not differentiable at 0 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 limit((f(x[0]+h)-f(x[0]))/h,h = 0,left) and limit((f(x[0]+h)-f(x[0]))/h,h = 0,right) exist, but

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limit((f(x[0]+h)-f(x[0]))/h,h = 0,left) <> limit((f... .

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For example : f(x) = abs(x) .

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f is continuous at 0, and

limit((abs(x)-0)/(x-0),x = 0,left) = -1

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limit((abs(x)-0)/(x-0),x = 0,right) = 1 .

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f is not differentiable at 0 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 limit((f(x[0]+h)-f(x[0]))/h,h = 0,left) or limit((f(x[0]+h)-f(x[0]))/h,h = 0,right) does not exist.

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For example : f(x) = x*sin(Pi/x) , for all x <> 0 , and f(0) = 0 .

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f is continuous at 0, and limit((x*sin(Pi/x)-0)/(x-0),x = 0,right) = limit(si... 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 -1 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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