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Continuous Functions

Key Questions

  • You can prove that a function #f(x)# is continuous at #a# by verifying that #lim_{x to a}f(x)=f(a)#.

  • No, it is not continuous at #x=3# since it is not defined there (zero denominator).

  • Answer:

    We may also state two alternative definitions of continuous functions, using either the sequential criterion or else using topology and open sets.

    Explanation:

    Alternative definition number 1
    Let #f: X ->Y# be a function and let #(x_n)# be a sequence in X converging to an element x in X, ie #lim(x_n)=x in X#
    Then f is continuous at x iff and only if the sequence of function values converge to the image of x undr f, ie #iff lim (f(x_n))=f(x) in Y#

    Alternative definition number 2
    Let #f: X ->Y# be a function. Then f is continuous if the inverse image maps open subsets of Y into open subsets in X.
    ie, #AAA_(open)subeY =>f^(-1)(A) # is open in X

Questions