Transcription of CONTINUITY AND DIFFERENTIABILITY
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CONTINUITY AND DIFFERENTIABILITY
Of a function at a pointLet f be a real function on a subset of the real numbers and let c be a point in thedomain of f. Then f is continuous at c iflim()()xcfxfc =More elaborately, if the left hand limit, right hand limit and the value of the functionat x = c exist and are equal to each other, ,lim()()lim()xcxcfxfcfx + ==then f is said to be continuous at x = in an interval(i)f is said to be continuous in an open interval (a, b) if it is continuous at everypoint in this interval.(ii)f is said to be continuous in the closed interval [a, b] if f is continuous in (a, b) limxa+ f (x) = f (a) limxb f (x) = f (b)Chapter5 CONTINUITY ANDDIFFERENTIABILITY20/04/2018 CONTINUITY AND DIFFERENTIABILITY meaning of CONTINUITY (i)Function f will be continuous at x = c if there is no break in the graph of thefunction at the point (),()cfc.(ii)In an interval, function is said to be continuous if there is no break in thegraph of the function in the entire function f will be discontinuous at x = a in any of the following cases :(i)limxa f (x) and limxa+ f (x) exist but are not equal.
CONTINUITY AND DIFFERENTIABILITY 87 5.1.3 Geometrical meaning of continuity (i) Function f will be continuous at x = c if there is no break in the graph of the function at the point (cfc, ()). (ii) In an interval, function is said to be continuous if there is no break in the
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