#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 $g(x)=\begin{cases} (x + 4) & \text{if } x < 0 \\ (x - 2)^2 & \text{if } x \ge 0 \end{cases}$ is not continuous at $x=0$.

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 $x=1$.

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Practice Question

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

#

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.