## An introduction to the theory of functions of a real variable |

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a+ib an—a an(x an+1 called choose closed interval a,b complex number contradiction converges to zero cosec coshx cosx decreasing defined denote the proposition denotes a number domain equation evaluate Examples expa expx f(xn finite set greatest lower bound greatest member greatest multiple Hence hypothesis identically zero Iff(x increasing interval 0,1 inverse inverse function least upper bound left limit Let f(x Let xn Mean Value Theorem means natural number non-decreasing numbers which satisfy off(x open interval oscillation partial fractions pn is true pn+1 is true polynomial polynomial h(x positive integer positive number prime factors Proof real number right limit Rolle's Theorem sequence set of numbers set of values sinh sinhx sinx sn(x sub-intervals sub-sequence sub-set suffices to prove suppose symbol takes the value thenf(x Uniform convergence unly value of f(x write P(x