Function l(x) is continuous for all real values of x and therefore has no point of discontinuity. Hence lim l(x) as x approaches -4 = 1 = l(-4). A function is known as a continuous function if it is. Function h is discontinuous at x = 1 and x = -1.ĭ) tan(x) is undefined for all values of x such that x = π/2 + k π, where k is any integer (k = 0, -1, 1, -2, 2.) and is therefore discontinuous for these same values of x.Į) The denominator of function j(x) is equal to 0 for x such that cos(x) - 1 = 0 or x = k (2 π), where k is any integer and therefore this function is undefined and therefore discontinuous for all these same values of x.į) Function k(x) is defined as the ratio of two continuous functions (with denominator x 2 + 5 never equal to 0), is defined for all real values of x and therefore has no point of discontinuity. In mathematics, continuity is a property of a function that decides its behaviour on a specific domain. The denominator is equal to 0 for x = 1 and x = -1 values for which the function is undefined and has no limits. Function g(x) is not continuous at x = 2.Ĭ) The denominator of function h(x) can be factored as follows: x 2 -1 = (x - 1)(x + 1). Therefore function f(x) is discontinuous at x = 0.ī) For x = 2 the denominator of function g(x) is equal to 0 and function g(x) not defined at x = 2 and it has no limit. Ziemer, “Weakly Differentiable Functions”, Springer-Verlag, Berlin, 1989.A) For x = 0, the denominator of function f(x) is equal to 0 and f(x) is not defined and does not have a limit at x = 0. Stampacchia, On some regular multiple integral problems in the calculus of variations, Comm. Morrey, “Multiple Integrals in the Calculus of Variations”, Springer-Verlag, New York, 1966. Miranda, Un teorema di esistenza e unicità per il problema dell'area minima in n variabili, Ann. Treu, Gradient maximum principle for minima, J. ![]() Treu, Existence and Lipschitz regularity for minima, Proc. Marcellini, Regularity for some scalar variational problems under general growth conditions, J. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. Hartman, On the bounded slope condition, Pacific J. Giusti, “Direct Methods in the Calculus of Variations” World Scientific, Singapore, 2003. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer-Verlag, Berlin, 1998. and then do a bunch of examples to see what is or isnt continuous using the. Giaquinta, “Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems”, Princeton University Press, Princeton, N.J., 1983. Continuity: Definition Example Meaning Concept Mathematics. ![]() Gariepy, “Measure Theorey and Fine Properties of Functions”, CRC Press, Boca Raton, FL, 1992. De Arcangelis, Some remarks on the identity between a variational integral and its relaxed functional, Ann. Notation Induction Logical Sets Word Problems. Wolenski, “Nonsmooth Analysis and Control Theory”, Graduate Texts in Mathematics, vol. Free function continuity calculator - find whether a function is continuous step-by-step. Sinestrari, “Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control”, Birkhäuser, Boston, 2004. Belloni, A survey on old and recent results about the gap phenomenon, In: “Recent Developments in Well-Posed Variational Problems”, R. ![]() Clarke, Local Lipschitz continuity of solutions to a basic problem in the calculus of variations, to appear. Bousquet, On the lower bounded slope condition, to appear.
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