Nondifferentiable FunctionsDate: Sun, 6 Nov 1994 14:24:49 -0500 (EST) From: Peter Finin Subject: Non differentiable function. In my calculus book, it mentions a function that is not differentiable at any point due to the fact that it is not smooth at any point. It does not go any farther than this, and I was interested in hearing more about this. Thanx. -Peter Finin From: Dr. Ken Date: Sun, 6 Nov 1994 14:53:24 -0500 (EST) Peter! Great question! If what you want to see is an example of a nasty function like that, I can give you one. In particular, let f(x) be a function that gives 0 as its output when x is rational, and 1 as its output when x is irrational. Since you're taking Calculus, I'll assume that you know about irrational and rational numbers, particularly that between any two irrational numbers is a rational number, and that between any two rational numbers is an irrational number. So the graph of this function looks kind of like a couple of blurry lines (that's how I think of it, anyway. It's not very technically precise, but I think it helps get an idea of how it looks), one at height zero and one at height one. Quite a jagged function. Is it continuous anywhere? Here's another one for you: F(x) = 0 when x is rational, and x when x is irrational. What are its properties? Where is it continuous? What does it look like? Where is it differentiable? -Ken |
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