Definition Method

Apply the definition of continuity directly to make the determination.

Determination Steps

1. Check whether the function is defined at that point

   • If the function is undefined at that point, it is discontinuous there

2. Compute the limit of the function at that point

   • Compute $\lim_{x \to x_0} f(x)$
   • If the limit does not exist, the function is discontinuous at that point

3. Compare the limit with the function value

   • If $\lim_{x \to x_0} f(x) = f(x_0)$, the function is continuous at that point
   • If $\lim_{x \to x_0} f(x) \neq f(x_0)$, the function is discontinuous at that point

Applied Examples

Example 1: determine the continuity of the function $f(x) = x^2 + 1$ at $x = 2$

Solution:

1. The function is defined at $x = 2$: $f(2) = 2^2 + 1 = 5$
2. Compute the limit: $\lim_{x \to 2} (x^2 + 1) = 2^2 + 1 = 5$
3. Compare: $\lim_{x \to 2} f(x) = f(2) = 5$
4. Conclusion: the function is continuous at $x = 2$

Example 2: determine the continuity of the function $f(x) = \frac{1}{x}$ at $x = 0$

Solution:

1. The function is undefined at $x = 0$
2. Conclusion: the function is discontinuous at $x = 0$

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Exercises

Exercise 1

Use the definition method to determine the continuity of the function $f(x) = x^3$ at $x = 1$.

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Summary

Symbols Used in This Article