What makes a function continuous at a point?

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embraci4i · Accepted Answer

Let             f              (        x        )             be a function defined in an interval             (      a      ,      b      )       and             x      0        ∈          (      a      ,      b      )       a point of the interval.Then the definition of continuity is that the limit of             f              (        x        )             as x approaches             x      0       equals the value of             f              (        x        )             in             x      0      .In symbols:            lim              x        →                  x          0                            f              (        x        )              =          f              (                  x          0                )            Based on the formal definition of limit, then, for every number       ε    &amp;gt;    0   we can find             δ      ε        &amp;gt;    0   such that:            |              x        -                  x          0                    |        &amp;lt;          δ      ε        ⇒          |                        f                      (            x            )                          -                  f                      (                          x              0                        )                              |        &amp;lt;    ε  This means that as x gets closer and closer to             x      0       also             f              (        x        )             gets closer and closer to             f              (                  x          0                )             and thus the function is &quot;smooth&quot;.

What makes a function continuous at a point?

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