Question

What is continuity at a point?

Answer 1

Let `f(x)` be a real function of real variable defined in the domain `I sub RR`.

Let `x_0 in I` be a point of accumulation for `I`.

The function `f(x)` is continuous in `x_0` if:

`lim_(x->x_0) f(x) = f(x_0)`

that is if the limit of `f(x)` as `x` approaches `x_0` equals the value of the function in `x_0`. This means that as `x` gets closer and closer to `x_0`, then `f(x)` gets closer and closer to `f(x_0)`, so that the graph of the function has no "jumps".