**Continuity**

A function is continuous in its domain if its graph is a solid unbroken line. All polynomial functions are continuous. Mathematically, (below) a function is defined to be continuous in *b* if the value of the function is equal to the limit of the function at this point.

**Example 1**

Determine the value of the function in* x = 2* such that the function is continuous.

**A piecewise function **

**Graph**

We find one-sided limits

The limit values are equal, so the function has a limit value in *x = 2*. The function is continuous in *x = 2* when determining the value of the function

**Example 2**

Let us examine whether the function *f* is continuous in *4*

A function is continuous in *4* if its limit value is equal to the value of the function at this point. A limit value exists if its one-sided limit values are equal.

One-sided limits

The value of the function in *4* is determined by the lower expression

The limit value of the function is equal to the value of the function in *4*, so the function is continuous at this point.

**Graph of ****f**

**Example 3**

Let us examine whether the function *f* is continuous in *5*

One-sided limits

The one-sided limit values are different, so the function does not have a limit value in *5*. Thus, the function is not continuous in *5*.

**Graph of ****f**

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