Calculus - Continuity and Differentiability

Continuity at a Point: A function $f(x)$ is continuous at $x = c$ if the limit of the function as $x$ approaches $c$ equals the value of the function at $c$. Visually, this means there is no break, hole, or jump in the graph at that point; you can draw the curve through $x = c$ without lifting your pencil from the paper.

Types of Discontinuity: Discontinuities are visual interruptions in a graph. A 'Removable Discontinuity' appears as a tiny hole on a smooth line. A 'Jump Discontinuity' occurs when the graph breaks and restarts at a different height (common in piecewise functions). An 'Infinite Discontinuity' occurs when the function approaches a vertical asymptote, shooting off towards $\infty$ or $-\infty$.

Continuity on an Interval: A function is continuous on a closed interval $[a, b]$ if it is continuous at every point in the open interval $(a, b)$, and the one-sided limits at the endpoints match the function values: $\lim_{x \to a^+} f(x) = f(a)$ and $\lim_{x \to b^-} f(x) = f(b)$.

Differentiability: A function is differentiable at $x = c$ if the derivative exists at that point, meaning the left-hand derivative equals the right-hand derivative. Visually, a differentiable graph is 'smooth' and has a unique tangent line. If a graph has a 'sharp corner' or 'cusp' (like the point of a 'V'), it is not differentiable there because no single tangent can be defined.

Relationship between Continuity and Differentiability: Every function that is differentiable at a point is necessarily continuous at that point. However, the reverse is not always true. Visually, a continuous curve might have a sharp point (like $y = |x|$ at $x=0$) where it is not differentiable, even though the line is never broken.

Intermediate Value Theorem (IVT): If a function $f$ is continuous on $[a, b]$, it must take every value between $f(a)$ and $f(b)$ at least once. Visually, a continuous path connecting two points at different heights must cross every horizontal line between those two heights.

Rolle's Theorem and Mean Value Theorem (MVT): Rolle's Theorem states that if a function is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, there is a point where the tangent is horizontal (slope = 0). Lagrange's MVT generalizes this, stating there is a point $c$ where the instantaneous slope (tangent) equals the average slope (secant) of the interval. Visually, at some point $c$, the tangent line is parallel to the chord joining the endpoints $(a, f(a))$ and $(b, f(b))$.