1. Local Linearity

2. A nice graph
Consider a small portion of the graph . . .

3. “Zooming In” Consider a small portion of the graph . . .
shown here in blue.

4. Zooming
Now “zoom in” on the blue part of the graph. . .

5. “Zooming In”
And repeat the process by zooming in on the part colored in pink. . .

6. “Zooming In”
Keep it up. . .

7. “Zooming In”

8. “Zooming In”

9. “Zooming In”

10. Typical Behavior

11. In general. . .
When we zoom in on a “sufficiently nice” function, we see a straight line.

12. Local Linearity

13. Differentiability