Continuity – AP Calculus

Intuitively, a function is continuous if you can draw its graph without lifting your pencil. There are no gaps, jumps, or holes. Most functions you work with in calculus (polynomials, trig functions, exponentials, logarithms) are continuous on their domains — but we need a precise definition to handle edge cases.

\text{A function } f \text{ is continuous at } x = a \text{ if:} $$ $$ \boxed{\lim_{x \to a} f(x) = f(a)}

📌 The Three Conditions for Continuity at a Point

A function $f$ is continuous at $x = a$ if and only if all three of these are true:

$f(a)$ is defined — the function has a value at $x = a$
$\displaystyle \lim_{x \to a} f(x)$ exists — the left and right limits agree (finite)
$\displaystyle \lim_{x \to a} f(x) = f(a)$ — the limit equals the function value

📊 Visualizing Continuity vs. Discontinuity

Use the interactive graph below to see continuous and discontinuous functions:

🚗 Real-World Analogy: A Smooth Road

Imagine driving on a road:

Continuous: A smooth, unbroken road — you never need to jump or teleport.
Removable discontinuity (hole): A small pothole at one point — you can drive around it, but the road technically has a gap at that exact spot.
Jump discontinuity: A sudden step or cliff — you'd need to jump to continue.
Infinite discontinuity (asymptote): A bottomless pit — you cannot cross at all.

📋 Types of Discontinuities

Type	Description	Condition	Example
Removable	A hole in the graph	$\lim_{x \to a} f(x)$ exists but $f(a)$ is undefined or not equal to the limit	$f(x)=\frac{x^2-1}{x-1}$ at $x=1$
Jump	Left and right limits exist but are different	$\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$	$f(x)=\lfloor x \rfloor$ (greatest integer)
Infinite	The function goes to $\pm\infty$	$\lim_{x \to a} f(x) = \infty$ or $-\infty$	$f(x)=\frac{1}{x}$ at $x=0$

Example 1Checking Continuity at a Point

Check if $f(x) = x^2$ is continuous at $x = 2$.

$f(2) = 4$ ✓ (defined)
$\lim_{x \to 2} x^2 = 4$ ✓ (limit exists)
$\lim_{x \to 2} f(x) = f(2)$ ✓ (they match)

All three conditions hold — $f$ is continuous at $x = 2$.

Example 2Removable Discontinuity (Hole)

Check continuity of $f(x) = \frac{x^2 - 4}{x - 2}$ at $x = 2$.

$f(2)$ is undefined (division by zero) → fails condition 1
However, $\lim_{x \to 2} \frac{x^2-4}{x-2} = \lim_{x \to 2} (x+2) = 4$
So $f$ has a removable discontinuity at $x = 2$

Example 3Jump Discontinuity

Check continuity of $f(x) = \lfloor x \rfloor$ (greatest integer function) at $x = 1$.

$\lim_{x \to 1^-} \lfloor x \rfloor = 0$
$\lim_{x \to 1^+} \lfloor x \rfloor = 1$
Since left ≠ right, $\lim_{x \to 1} f(x)$ does not exist → jump discontinuity

Example 4Infinite Discontinuity (Vertical Asymptote)

Check continuity of $f(x) = \frac{1}{x}$ at $x = 0$.

$f(0)$ is undefined → fails condition 1
$\lim_{x \to 0} \frac{1}{x}$ does not exist (goes to $\pm\infty$)
So $f$ has an infinite discontinuity (vertical asymptote) at $x = 0$

📏 Continuity on an Interval

A function is continuous on an interval if it is continuous at every point in that interval. For example:

All polynomials are continuous on $(-\infty, \infty)$
Rational functions are continuous on their domains (everywhere except where denominator = 0)
$\sin x$ and $\cos x$ are continuous on $(-\infty, \infty)$
$\tan x$ is continuous on intervals that avoid $x = \frac{\pi}{2} + n\pi$

⚠️ Common Mistakes

Assuming a function is discontinuous just because it has a hole — correct, but the limit still exists (removable).
Confusing "limit exists" with "function is continuous" — continuity requires the limit to also equal the function value.
Thinking all piecewise functions are discontinuous — many are continuous if the pieces match at the boundaries.

🔍 Key Takeaways

A function is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$.
Three conditions: (1) $f(a)$ defined, (2) limit exists, (3) limit equals $f(a)$.
Holes are removable discontinuities (limit exists, function undefined or mismatch).
Jumps occur when left and right limits differ.
Infinite discontinuities occur when the function goes to $\pm\infty$ (vertical asymptotes).

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