For a function f : [ − 1 , 1 ] → R, consider the following statements:Statement 1: If ‘ lim n → ∞ f ( 1 n ) = f ( 0 ) = lim n → ∞ f ( − 1 n ) ‘ ,then f is continuous at x = 0Statement 2: If f is continuous at x = 0, then lim n → ∞ f ( 1 n ) = lim n → ∞ f ( − 1 n ) = lim n → ∞ f ( e 1 n − 1 ) = f ( 0 )Then which of the above statements is/are true.My Attempt:I feel that lim n → ∞ f ( 1 n ) = f ( 0 ) is same as lim x → 0 f ( x ) = f ( 0 ), so f should be continuous at x = 0. So statement 1 must be true.In statement 2, since f is continuous at x = 0 we have lim n → ∞ f ( 1 n ) = f ( lim n → ∞ 1 n ) = f ( 0 ). By same logic we can prove lim n → ∞ f ( − 1 n ) = lim n → ∞ f ( e 1 n − 1 ) = f ( 0 )Can there be counter-examples to what I am thinking

Question

For a function   f  :  [  −  1  ,  1  ]  →  R, consider the following statements:Statement 1: If  ‘      lim          n      →      ∞        f      (          1      n        )    =  f  (  0  )  =      lim          n      →      ∞        f      (    −          1      n        )    ‘  ,then   f is continuous at   x  =  0Statement 2: If   f is continuous at   x  =  0, then      lim          n      →      ∞        f      (          1      n        )    =      lim          n      →      ∞        f      (    −          1      n        )    =      lim          n      →      ∞        f      (          e                        1          n                      −    1    )    =  f  (  0  )Then which of the above statements is/are true.My Attempt:I feel that       lim          n      →      ∞        f      (          1      n        )    =  f  (  0  ) is same as       lim          x      →      0        f  (  x  )  =  f  (  0  ), so   f should be continuous at   x  =  0. So statement 1 must be true.In statement 2, since   f is continuous at   x  =  0 we have       lim          n      →      ∞        f      (          1      n        )    =  f      (          lim              n        →        ∞                    1      n        )    =  f  (  0  ). By same logic we can prove       lim          n      →      ∞        f      (    −          1      n        )    =      lim          n      →      ∞        f      (          e                        1          n                      −    1    )    =  f  (  0  )Can there be counter-examples to what I am thinking

Helena Ward · Accepted Answer

For first statement  f  (  x  )  =      {                            0                     if           x          =                      1            n                              n          ∈                      Z                              ∖                    {          0          }                                                          0                              if           x          =          0                                                          1                     otherwise                        So      lim          n      →      ∞        f      (          1      n        )    =      lim          n      →      ∞        f      (    −          1      n        )    =  0  =  f  (  0  )But      lim          n      →      ∞        f      (                  3            n        )    =  1Hence for every   lim      x    n    =  0 our   lim  f  (      x    n    )  ≠  f  (  0  )Hence statement is falseSecond statement is obvious becouse of continuity of function

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