Geometrical meaning of continuity
Geometrical Meaning of Continuity
Definition:
 Geometrically Continuous Function: A function $f(x)$ is considered geometrically continuous if its graph can be drawn without lifting the pen from the paper within a specific interval or at a particular point.
Graphical Interpretation:

Smoothness of Graph: A continuous function has a smooth, unbroken graph without jumps, breaks, or abrupt changes.

Connectedness: The graph of a continuous function remains connected without any gaps or disruptions in its curve.
Characteristics of Continuous Functions:

No Sudden Jumps: The function's graph doesn't have sudden vertical jumps; it transitions smoothly between points.

No Holes or Gaps: There are no gaps or holes in the graph that would require lifting the pen to draw the curve.
Visualizing Discontinuities:

Removable Discontinuity: A hole in the graph where the function could be redefined to make it continuous.

Jump Discontinuity: The graph jumps from one point to another, indicating a sudden change in function values.

Infinite Discontinuity: The function has vertical asymptotes, causing an unbounded increase or decrease in function values.
Example of Geometric Continuity:
Consider the function $f(x)=\frac{{x}^{2}1}{x1}$, which is not defined at $x=1$.
 Geometrically: The graph of this function has a hole at $x=1$ because it's undefined there.
Understanding via Graph:

Graph Behavior: If you were to plot $f(x)=\frac{{x}^{2}1}{x1}$, you'd observe a gap at $x=1$, indicating a removable discontinuity.

Approaching the Gap: As $x$ approaches 1 from both sides, the graph's values get closer to the same value (which would fill the gap if defined).