Continuity

$$\lim_{x\to{c}}f(x) = f(c)$$

In words:

A function $f(x)$ is continuous at a point $c$ if the limit of the function approaching that point ($\lim_{x\to{c}}f(x)$) is equal to the value of the function at that point ($f(c)$)

A Sample

Let us define following piecewise function

$${}$$

The limit $L$ of the piecewise function for $x\to{0}$ is $0$

$${}$$

You can use the following slider to regulate the function value at 0

You can use the following slider to regulate $\epsilon$

You can use the following slider to regulate $\delta$

Notice how:

  1. If you use the sliders for $\epsilon$ and $\delta$, you can find that the limit of the piecewise function is $0$. This is independant of the value at $0$ itself.
  2. If you use the slider to change the value at zero and you set it to $0$, then the piecewise function is continuous: the value at $0$ is equal to the limit when approaching $0$.
  3. If you use the slider to change the value at zero and you set it to something different than $0$, then the piecewise function is not continuous: the value at $0$ is different of the limit when approaching $0$.