In single variable calculus for f ( x ) to be continuous at x 0 we need to have lim h → 0 f ( x 0 − h ) = lim h → 0 f ( x 0 + h ) = lim h → 0 f ( x 0 ) and it should be well defined at that point.

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In single variable calculus for   f  (  x  ) to be continuous at       x    0   we need to have       lim          h      →      0        f  (      x    0    −  h  )  =      lim          h      →      0        f  (      x    0    +  h  )  =      lim          h      →      0        f  (      x    0    ) and it should be well defined at that point.

Bernard Walker · Accepted Answer

For   f  :            R        n    →      R  , to prove that   f is continuous at             x                    x        0  , it is sufficient to show that   f  (            x              x    )  →  f  (            x                    x        0    ) whenever   d  (            x              x    ,            x                    x        0    )  →  0, where   d  :            R        n    ×            R        n    →      R   is any continuous function such that   d  (      x    1    ,  .  .  .  ,      x    n    ,            x                    x        0    )  →  0 forces       x    1    ,  .  .  .  ,      x    n   to go simultaneously to the corresponding components of             x                    x        0  . Convenient forms of such   f are   f  (            x              x    ,            y              y    )  =      |        x    1    −      y    1        |    +  ⋯  +      |        x    n    −      y    n        |   or   f  (            x              x    ,            y              y    )  =  (      x    1    −      y    1        )    2    +  ⋯  +  (      x    n    −      y    n        )    2  .

In single variable calculus for f(x) to be continuous at x_0 we need to have lim_(h->0) f(x_0−h)=lim_(h->0)f(x_0+h)=lim_(h->0) f(x_0) and it should be well defined at that point.

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