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continuity

Shows continuity at x=a as a 3-item checklist: (1) f(a) is defined (filled point), (2) the limit exists (left/right approach markers agree), and (3) the limit equals the function value. The animation cycles through broken cases (a hole or a jump) and then smoothly fixes them so lim_{x->a} f(x)=f(a).

canvasclick to interact
t=0s

practical uses

  • 01.Diagnose discontinuities in piecewise functions (holes vs jumps).
  • 02.Understand when limit computations actually tell you the function’s value.
  • 03.Justify when calculus rules (like plugging in for polynomials) apply because the function is continuous.

technical notes

Pure Canvas2D. Uses a snapped grid for a blocky aesthetic, time-based phases to highlight the three continuity conditions, and alternates between removable/jump discontinuities before easing into a corrected continuous function. Responsive scaling via scale = min(w,h)/baseSize.