#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 $x$ ...