MTH 251 – Differential Calculus Chapter 2 – Limits and ContinuityIntuitive Concepts …Local DiscontinuityLocal Continuity – DefinedSlide 5Types of Local DiscontinuitySlide 7Slide 8Continuous FunctionsContinuous Functions – ExamplesLimits of Compositions of Continuous FunctionsThe Intermediate Value TheoremApplication of the IVTMTH 251 – Differential CalculusChapter 2 – Limits and ContinuitySection 2.5ContinuityCopyright © 2010 by Ron Wallace, all rights reserved.Intuitive Concepts …•What would it mean for a curve to be “unbroken”?•Is “unbrokenness” a local concept or global concept?•This section … give a precise mathematical definition to the characteristics of “unbroken” as applied to functions.•Mathematically, this concept is called …continuityLocal DiscontinuitycUndefined PointcJumpcAsymptotecUniquely Defined PointLocal Continuity – Defined•Continuous at a PointInterior point of the domainLeft endpoint of the domainRight endpoint of the domainlim ( ) ( )x cf x f c�=lim ( ) ( )x cf x f c+�=lim ( ) ( )x cf x f c-�=Note that in all three cases, the limit must exist and the function must be defined at the point.Local DiscontinuityccccUndefined PointJumpAsymptoteUniquely Defined Pointf(c) is not defined.The limit does not exist.The limit and f(c) are not equal.f(c) is not defined and the limit does not exist.Types of Local DiscontinuityccccUndefined PointJumpAsymptoteUniquely Defined Pointf(c) is not defined.The limit does not exist.The limit and f(c) are not equal.f(c) is not defined and the limit does not exist.RemovableJumpInfiniteRemovableTypes of Local Discontinuity•Removable – hole or uniquely defined pointIf the value at that point is changed to the correct value, the function would become continuous (this new function is called the “continuous extension” of the original function).•Example:The following function is not continuous at x=1Find the continuous extension321( )1xf xx-=-Types of Local Discontinuity•Removable – hole or uniquely defined pointIf the value at that point is changed to the correct value, the function would become continuous (this new function is called the “continuous extension” of the original function).•Jump – piecewise functions where the pieces do not connect.•Infinite – a vertical asymptote•Oscillating – oscillates between values1( ) cosf xx� �=� �� �01lim cos ?xx�� �=� �� �Continuous Functions•Continuous on an IntervalThe function is continuous for all values in the interval.•Continuous FunctionThe function is continuous for all values of its domain (i.e. f(c) existing is a given).•NOTE: “Continuous Function” does not necessarily mean continuous at all real numbers (i.e. everywhere).Example: 1( )xf x =Continuous Functions – Examples•Polynomial Functions•Rational Functions (not where the denominator is zero)•Absolute Value Functions•Radical Functions (domain may be limited)•Trigonometric Functions and their Inverses (domain may be limited)•Exponential and Logarithmic Functions (domain may be limited)•Combinations of Continuous Functionssums, differences, products, multiples, quotients, powers, roots, & compositionsNote: domain may change or be limitedLimits of Compositions of Continuous Functions•If g(x) is continuous at f(c), then …•Example:( )( )( )lim lim ( )x c x cg f x g f x� �=( )3lim sin 2 5 xx�- =( )( )3sin lim 2 5 xx�- =The Intermediate Value Theorem•If f(x) is continuous over [a,b] and k is any value between f(a) and f(b), then there exists at least one number c in [a,b] where f(c) = kf(x)abf(a)f(b)kc1c2c3Application of the IVT•Finding roots of functions.If f(x) is continuous over [a,b] and f(a) and f(b) have opposite signs, then there is a root between a and b.•Example … Show that there is a root between 3 & 4 for the function f(x) = x2 – 2x –