What is Continuity?

All three must be true! If even one fails, the function is discontinuous at that point.

Testing for Continuity

Example 1: Is $f(x) = x^2$ continuous at x = 3?

Check 1: Does f(3) exist? $f(3) = 3^2 = 9$ ✓ Yes

Check 2: Does $\lim_{x \to 3} f(x)$ exist? $\lim_{x \to 3} x^2 = 9$ ✓ Yes

Check 3: Does $\lim_{x \to 3} f(x) = f(3)$? $9 = 9$ ✓ Yes

Conclusion: f is continuous at x = 3!

Example 2: Not Continuous

$f(x) = \begin{cases} x + 1 & \text{if } x \neq 2 \\ 5 & \text{if } x = 2 \end{cases}$

Is this continuous at x = 2?

Check 1: Does f(2) exist? $f(2) = 5$ ✓ Yes

Check 2: Does $\lim_{x \to 2} f(x)$ exist? $\lim_{x \to 2} (x + 1) = 3$ ✓ Yes

Check 3: Does $\lim_{x \to 2} f(x) = f(2)$? $3 \neq 5$ ✗ NO!

Conclusion: f is NOT continuous at x = 2 (even though f(2) exists and the limit exists!)

Continuous Everywhere

Many familiar functions are continuous everywhere:

• Polynomials: $x^2$, $x^3 - 2x + 1$, etc.
• Exponential functions: $e^x$, $2^x$
• Sine and cosine: $\sin(x)$, $\cos(x)$
• Square root (on its domain): $\sqrt{x}$ for $x \geq 0$

Common Discontinuities

Functions are typically discontinuous where:

• Rational functions have division by zero
• Piecewise functions change formulas
• Absolute value creates a corner (still continuous, but not differentiable!)

Why Continuity Matters

Continuous functions have nice properties:

• Intermediate Value Theorem: If continuous on [a, b], it takes every value between f(a) and f(b)
• Can find limits by direct substitution
• Behave predictably - no surprises!

Visual Test

On a graph, a function is continuous at a point if:

• No gap (both sides connect)
• No hole (no open circle)
• No jump (no sudden leap)
• The point is actually on the curve

Find the limit from the left: $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x^2 = 4$

Find the limit from the right: $\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (kx + 1) = 2k + 1$

For continuity, these must all be equal: $4 = 2k + 1$ $3 = 2k$ $k = \frac{3}{2}$

Answer: $k = \frac{3}{2}$

With this value, f(2) = 4 and both one-sided limits equal 4, making the function continuous!