Continuity at a Point

Table of Contents

1. Definition of Continuity
2. Formal Definition of Continuity at a Point
3. Examples of Continuity and Discontinuity
4. Continuity in Piecewise-Defined Functions
5. Continuity in Combinations of Functions
6. Worked Examples
7. Continuity Over an Interval
8. Summary and Key Takeaways
9. Glossary
10. Practice Questions

Definition of Continuity

What does continuous mean in the context of functions?

A function is continuous if you can draw its graph without lifting your pencil off the paper. This intuitive idea is used to help understand the behavior of functions in calculus.

Formal Definition of Continuity at a Point

What is the definition for a function to be continuous at a point?

A function $f$ is continuous at the point $x=c$ if:

1. $f(c)$ exists,
2. $\lim_{{x \to c}} f(x)$ exists,
3. $\lim_{{x \to c}} f(x) = f(c)$.

In simpler terms:

• The function must have a well-defined finite value at $x=c$.
• The function must have a well-defined finite limit at $x=c$.
  • This means that the one-sided limits must be equal at $x=c$.
• The value of the limit and the value of the function must be equal at $x=c$.

Examples of Continuity and Discontinuity

Consider the functions $f$, $g$ and $h$ defined by:

• $f(x) = \frac{1}{x}$
• $g(x) = \frac{x}{x}$
• h(x) = \begin{cases} x & \text{if } x \ne 0\\ 1 & \text{if } x=0 \end{cases}

For function $f$:

• $\frac{1}{0}$ is not defined, so $f(0)$ does not exist.
• $f$ becomes unbounded as $x$ approaches zero from the left and right, so $\lim_{{x \to 0}} f(x)$ doesn't exist.
• Therefore, $f$ is not continuous at $x=0$.

For function $g$:

• $\lim_{{x \to 0^{-}}} g(x) = \lim_{{x \to 0^{+}}} g(x) = 1$, so $\lim_{{x \to 0}} g(x) = 1$.
• But $\frac{0}{0}$ is not defined, so $g(0)$ doesn't exist.
• Therefore, $g$ is not continuous at $x=0$.

For function $h$:

• $h(0) = 1$, so the function has a well-defined finite value at $x=0$.
• $\lim_{{x \to 0^{-}}} h(x) = \lim_{{x \to 0^{+}}} h(x) = 0$, so $\lim_{{x \to 0}} h(x) = 0$.
• But $\lim_{{x \to 0}} h(x) \ne h(0)$.
• Therefore, $h$ is not continuous at $x=0$.

Continuity in Piecewise-Defined Functions

How does continuity work for a piecewise-defined function?

The definition of continuity at a point applies to piecewise-defined functions. In practical terms:

• At a boundary to a partition of the function's domain, a piecewise-defined function will be continuous if the following are all equal:
  • The value of the expression defining the function to the left of the boundary.
  • The value of the expression defining the function to the right of the boundary.
  • The value of the function at the boundary.

This is not an alternative definition of continuity but a consequence of the standard definition.

Continuity in Combinations of Functions

How does continuity work for combinations of functions?

You can use the continuity theorem for combinations of functions:

• If functions $f$ and $g$ are continuous at a point $x=a$, then the following functions are also continuous at $x=a$:
  • $f+g$
  • $f-g$
  • $f \cdot g$
  • $\frac{f}{g}$ (as long as $g(a) \ne 0$)

Worked Examples

Example 1

Problem: Explain why the function $f$ defined by $f(x) = \frac{x+3}{x-1}$ is not continuous at $x=1$.

Answer: The main problem here is that $f$ is not defined when $x=1$: $$f(1) = \frac{1+3}{1-1} = \frac{4}{0}$$ which is not defined.

Alternatively, you could show that the limits from the left and right are unbounded at $x=1$, which also makes the function discontinuous at that point.

Conclusion: $f$ is not continuous at $x=1$.

Example 2

Problem: Explain why the function $g$ defined by g(x)=\begin{cases} (x + 4) & \text{if } x < 0 \\ (x - 2)^2 & \text{if } x \ge 0 \end{cases}<math-inline>is not continuous at</math-inline>x=0.

Answer: First, look at the left and right limits at $x=0$: $$\lim_{{x \to 0^{-}}} g(x) = (0) + 4 = 4$$ $$\lim_{{x \to 0^{+}}} g(x) = (0 - 2)^2 = 4$$

Those are equal, so the limit exists: $$\lim_{{x \to 0}} g(x) = 4$$

Now, look at the value of the function at $x=0$: $$g(0) = 2$$

So, both the value of the function and the value of its limit are well-defined at $x=0$. The problem is that those two values are not equal.

Conclusion: $g$ is not continuous at $x=0$.

Continuity Over an Interval

What does it mean for a function to be continuous over an interval?

A function is continuous over an interval if it is continuous at every point in the interval.

• For example, we have seen that $g(x)=\frac{x}{x}$ is not continuous at $x=0$, but it is continuous over any interval that doesn't include 0. ### What does it mean for a function to be continuous?

A function is said to be continuous if it is continuous at every point in its domain.

• For example, we have seen that $f(x)=\frac{1}{x}$ is not continuous at $x=0$.
  • Therefore, the function $f$ is not continuous over all the real numbers.
• However, if we define the function $j$ as $j(x)=\frac{1}{x}$, $x \ne 0$,
  • We have removed the discontinuity from its domain.
  • Therefore, $j$ is a continuous function.

Summary and Key Takeaways

Summary

• Continuity of a function at a point requires that the function is defined at that point, the limit exists, and the function's value equals the limit at that point.
• The concept of continuity extends to piecewise-defined functions and combinations of functions.
• A function can be continuous over an interval if it is continuous at every point within that interval.

Key Takeaways

• Continuity at a Point: Requires $f(c)$ exists, $\lim_{{x \to c}} f(x)$ exists, and $\lim_{{x \to c}} f(x) = f(c)$.
• Piecewise Functions: Continuity at boundaries requires left-hand limit, right-hand limit, and function value to be equal.
• Combination of Functions: Sum, difference, product, and quotient (if the denominator is not zero) of continuous functions are also continuous.
• Interval Continuity: A function is continuous over an interval if it is continuous at every point in that interval.

Glossary

• Continuous Function: A function that is unbroken at a point or over an interval.
• Limit: The value that a function approaches as the input approaches some value.
• Piecewise-Defined Function: A function defined by different expressions over different parts of its domain.
• One-Sided Limit: The limit of a function as the input approaches a point from one side (left or right) only.
• Domain: The set of all possible input values for which a function is defined.

Practice Questions

Practice Question

Question 1: Determine if the function $f(x) = \frac{2x^2 - 4x + 2}{x-1}$ is continuous at $x=1$.

Practice Question

Question 2: Explain why the function h(x)=\begin{cases} 2x+1 & \text{if } x < 2 \\ 3x-2 & \text{if } x \ge 2 \end{cases}<math-inline>is or isn't continuous at</math-inline>x=2.

Practice Question

Question 3: For the function $k(x) = \frac{x^2 - 1}{x-1}$, find the limit as $x$ approaches 1 and determine if $k$ is continuous at $x=1$.

Conclusion

Understanding the concept of continuity is essential for mastering calculus. By practicing these definitions and examples, you can develop a strong foundation in recognizing and proving continuity for various functions.