Question

What makes a function continuous at a point?

Answer 1

Let `f(x)` be a function defined in an interval `(a,b)` and `x_0 in (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 `epsilon > 0` we can find `delta_epsilon > 0` such that:

`abs(x-x_0) < delta_epsilon => abs(f(x)-f(x_0)) < epsilon`

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 "smooth".