Continuity
Continuity means a function has no breaks, holes, or jumps and can be drawn without lifting your pencil.
Formula
\lim_{x \to a} f(x) = f(a)
Definition
A function is continuous if its graph has no holes, jumps, or gaps, meaning you can trace it with a pencil without lifting it off the paper. Formally, $f$ is continuous at $x = a$ if three conditions hold: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x) = f(a)$; equivalently, for every $\epsilon > 0$ there exists $\delta > 0$ such that $|x - a| < \delta$ implies $|f(x) - f(a)| < \epsilon$. Uniform continuity strengthens this by requiring $\delta$ to work globally, independent of $a$.
Example
The graph of $y = x^2$ is one smooth curve with no breaks, but $y = 1/x$ has a break at $x = 0$ because division by zero is undefined there. $f(x) = (x^2 - 1)/(x - 1)$ is not defined at $x = 1$ (a hole), so it is discontinuous there, though redefining $f(1) = 2$ makes it continuous (a removable discontinuity). $f(x) = x^2$ is continuous on all of $\mathbb{R}$ but not uniformly continuous, since $\delta$ depends on the point, while $f(x) = \sin(x)$ is uniformly continuous on $\mathbb{R}$ because its derivative is bounded.
Key Insight
Continuity is the mathematical way of saying a function behaves predictably with no sudden surprises. It underlies the Intermediate Value Theorem: a continuous function on $[a,b]$ hits every value between $f(a)$ and $f(b)$, which is how we know equations like $x^3 = 2$ must have a solution. Uniform continuity is the stronger condition that guarantees a function is Riemann integrable and that Cauchy sequences in the domain map to Cauchy sequences in the range.