MATA31H3 Lecture Notes - Lecture 8: Classification Of Discontinuities, Intermediate Value Theorem, Infimum And Supremum
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F (c) is defined (c is in the domain of f(x)) lim x!c lim x!c f (x)exists f (x) = f (c) 6. 1 continuity at a point and on an interval. Intuitively we thought of continuous functions as of functions with graphs we can sketch. Without picking up our pencil in this section we"ll develop a formal definition of continuity using limits. If function f defined on open interval (c ! p,c + p) , we say that f is continuous at c if lim x!c. Graph of continuous function f is continuous at every number in an interval (has no breaks in this interval). The above implies: f (x) = f (c) Function f (x) is discontinuous at c if one of the (1), (2), or (3) fails. The restriction 0 < x !c can be removed because if x = c, f (x) = f (c) then f (x) ! f (c) = 0.
