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>Limits>How Do You Determine Continuity of a Function?

How Do You Determine Continuity of a Function?

Continuity is a property of a function. When you talk about continuity, you describe whether the graph of a function exists for all values of $x$ in an interval, and that these points are adjacent to each other, meaning there are no gaps between any of them.

When a graph is continuous, it means that you can draw it without lifting your pencil. By contrast, there are graphs where the $y$ -value of a given $x$ -value is some distance away from the $y$ -value of an adjacent $x$ -value. These graphs are not continuous. Functions that are not defined for all $x$ -values on an interval are also not continuous on that interval. Below, you see drawings of the different cases:

Continuous function and discontinuous function

Continuous function and discontinuous function

Theory

Continuity

If $\lim_{x \to a} f (x) = f (a)$ , then $f$ is continuous for $x = a$ .
If $\lim_{x \to a} f (x) \neq f (a)$ , then $f$ is discontinuous for $x = a$ .

When

f (x)

is continuous for all

x

in an interval, we say that it is continuous on the interval.

Example 1

Determine whether the function $f (x) = \frac{x^{2} - 2 x}{x - 1}$ is continuous for $x = 1$

\begin{array}{l} \lim_{x \to 1} \frac{x^{2} - 2 x}{x - 1} & = \lim_{x \to 1} \frac{\frac{x^{2}}{x^{2}} - \frac{2 x}{x^{2}}}{\frac{x}{x^{2}} - \frac{1}{x^{2}}} \\ = \lim_{x \to 1} \frac{1 - \frac{2}{x}}{\frac{1}{x} - \frac{1}{x^{2}}} \\ = \lim_{x \to 1} \frac{1 - \frac{2}{1}}{\frac{1}{1} - \frac{1}{1^{2}}} \\ = \frac{1 - 2}{1 - 1} \\ = not defined \end{array}

You see that

f (x)

is discontinuous, since you got an invalid expression when calculating the limit.

Example 2

Determine if the function $f (x) = \frac{x^{2} - 2 x + 2}{x + 3}$ is continuous for $x = - 2$

\begin{matrix} \lim_{x \to - 2} \frac{x^{2} - 2 x + 2}{x + 3} \\ = \frac{{(- 2)}^{2} - 2 (- 2) + 2}{- 2 + 3} \\ = \frac{4 + 4 + 2}{1} \\ = 10 \end{matrix}

You see that $f (x)$ is continuous since you got a numeric value as the result when you calculated the limit.

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