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2.5 continuity - infinity(vertical asymptote One sided...

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2.5 Continuity Def: a function f(x) is Continuous at x = a Def: a function is continuous if a line can be drawn if you can trace it without picking up your pencil. This extends direct sub to other continuous functions This means 3 Things : 1. f(a) exists 2. Lim f(x) exists 3. they are the same We say f has a discontinuity at x = a, or f is discontinuous at x = a if f is no continuous at x = a Def: a function f(x) is continuous on the interval, I, if it is continuous at each x in I Discontinuities are one of 3 types 1. Removable – if I can define the function or redefine the function (hole) 2. jump – in order to finish the graph, it has to jump to a different spot (jump) 3. unbounded – when the function approaches positive or negative
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Unformatted text preview: infinity (vertical asymptote) One sided Continuity – one sided limit Theorem: f and g continuous implies 1. f + or – g continuous 2. f * g continuous 3. f/g or g/f continuous where defined 4. c*f continuous Theorem: 1. Polynomials are continuous everywhere 2. Rational functions are continuous where defined 3. Roots are continuous where defined 4. Trigs are continuous where defined Theorem: if f(x) is continuous then lim f(g(x)) is f(lim g(x)) The intermediate Value Theorem: Given: f(x) continuous on [a,b] and N between f(a) and f(b). Then: there exists a C so that a<c<b and f(c) = N...
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