Transcription of Differentiable Implies Continuous
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Differentiable Implies Continuous
Differentiable Implies Continuous Theorem: If f is Differentiable at x0, then f is Continuous at x0. We need to prove this theorem so that we can use it to find general formulas for products and quotients of functions. We begin by writing down what we need to prove; we choose this carefully to make the rest of the proof easier. We want to show that: lim f(x) f(x0) = 0. xx0 This is the same as saying that the function is Continuous , because to prove that a function was Continuous we d show that lim f(x) = f(x0). xx0 We prove lim f(x) f(x0) = 0 by multiplying and dividing it by the same xx0 number this won t change its value. lim f(x) f(x0) = lim f(x) f(x0)(x x0) xx0 xx0 x x0 = f (x) 0 =0.
Differentiable Implies Continuous Theorem: If f is differentiable at x 0, then f is continuous at x 0. We need to prove this theorem so that we can use it to find general formulas
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