Essential Knowledge 2.2B2

Essential Knowledge 2.2B2 Students will know that if a function is differentiable at a point, then it is continuous at that point.

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As you saw in the previous Essential Knowledge, continuity was a requirement for differentiability. This means that differentiability is a stricter requirement for a function than continuity. This means we can say

• all differentiable functions are continuous

• not all continuous functions are differentiable

Think of it like a square and a rectangle. All squares are rectangles, but not all rectangles are squares. This is because squares have to meet the same rules as rectangles, but also have additional rules which rectangles don't have to meet

We also have a third category for functions we have learned: the limit exists. This is the least strict category, because limits must exist everywhere for a function to be continuous, but continuity also requires that the limiting values equal the actual values at each point.

To sum it up

• "Limit Exists"

  • Least strict category

  • Limit of f(x) as x approaches a exists

• "Continuous"

  • Stricter than "Limit Exists"

  • Limit of f(x) as x approaches a exists AND limit of f(x) as x approaches a equals f(a)

• "Differentiable"

  • Stricter than "Continuous"

  • Continuous AND no corners (aka derivative from left equals derivative from right)