Continuity and Differentiability: common misconceptions

If your child says… “Any value of k works because we choose it separately.”

Ask together

If you assign a different value, will it match either approach?

What’s true

The assigned value must match what nearby values approach.

If your child says… “One counts as a discontinuity on this function’s domain.”

Ask together

Does 1 belong to the domain you wrote?

What’s true

Continuity on a domain checks only the inputs belonging to it.

If your child says… “There is no break, so there must be a derivative.”

Ask together

Can −1 and 1 give one common slope?

What’s true

A derivative guarantees continuity, but a continuous graph can still have unequal side slopes.

If your child says… “I only need to differentiate the fifth power.”

Ask together

Where would the factor from the inside expression’s rate appear in your answer?

What’s true

The outside rate must be multiplied by the inside rate.

If your child says… “The derivative of y squared is just twice y.”

Ask together

Which factor records how y changes when x changes?

What’s true

Differentiating with respect to x must include how y changes with x.

If your child says… “The logarithm of zero is zero.”

Ask together

Where does the exponential curve reach an output of zero?

What’s true

Every exponential output is positive, so its inverse logarithm accepts only positive inputs.

If your child says… “The expression beside y-prime divided by y is already the derivative.”

Ask together

What must you multiply by to leave y-prime alone?

What’s true

Division by y remains until you multiply both sides by y.

If your child says… “Differentiating the slope in t gives the second derivative with respect to x.”

Ask together

When t increases by 1, how much does x increase?

What’s true

The second derivative measures slope change per change in x.