Identify graphs of continuous functions

Learning Objectives Visually identify continuous and discontinuous functions from their graphs. Define continuity at a point using the three-part limit definition. Classify different types of discontinuities (removable, jump, infinite) by analyzing their graphical features. Identify the specific x-values where a function is discontinuous by inspecting its graph. Explain why a function is discontinuous at a specific point by referencing its graphical features like holes, jumps, and asymptotes. Connect the intuitive concept of 'drawing without lifting the pencil' to the formal definition of continuity. Determine the interval(s) over which a function is continuous based on its graph. Ever tried to trace a complex drawing without lifting your pencil? ✏️ You were actu...

TermDefinitionExample Continuous FunctionA function whose graph can be drawn as a single, unbroken curve without lifting your pencil.The graph of f(x) = x² is a smooth parabola that can be drawn in one continuous motion. Discontinuous FunctionA function whose graph has one or more breaks, gaps, or holes. It cannot be drawn in a single, unbroken motion.The graph of f(x) = 1/x has a break at x=0, where you must lift your pencil to draw the two separate branches. Removable DiscontinuityA single point missing from the graph, often represented as an open circle or 'hole'. The discontinuity can be 'removed' by defining the function at that single point.A graph that looks like the line y = x + 1 but has an open circle at the point (2, 3). Jump DiscontinuityA break in the grap...

The Three-Part Definition of Continuity at a Point A function f(x) is continuous at x = c if: 1. f(c) is defined. 2. \lim_{x \to c} f(x) exists. 3. \lim_{x \to c} f(x) = f(c). This is the formal test for continuity at a single point. Visually, this means: 1) There is a solid dot or a solid line at x=c. 2) The graph approaches the same y-value from both the left and the right. 3) The solid dot is exactly where the approaching lines meet. Condition for a Jump Discontinuity A jump discontinuity exists at x = c if \lim_{x \to c^-} f(x) ≠ \lim_{x \to c^+} f(x), but both one-sided limits exist as finite numbers. Use this to formally identify a jump. On a graph, this looks like a sudden vertical leap from one y-value to another. The graph approaches a different height from the left...