# Geometrical meaning of continuity

## Geometrical Meaning of Continuity

### Definition:

• Geometrically Continuous Function: A function $f\left(x\right)$ 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 $f\left(x\right)=\frac{{x}^{2}-1}{x-1}$, 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 $f\left(x\right)=\frac{{x}^{2}-1}{x-1}$, 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).