Formula sheet: Continuity and Differentiability

1.1 Continuity and differentiability tests

A function is continuous at a point when the left limit, the right limit and the value at the point all agree. A function is differentiable at a point when the left and right derivatives exist and are equal. Differentiability is the stronger condition: every differentiable function is continuous, but a continuous function need not be differentiable.

On an interval, a function is continuous when it is continuous at every interior point and one-sided continuous at the endpoints. Polynomials, sine, cosine, exponentials and logarithms are continuous on their domains.

1.2 Limits behind the definitions

Both definitions are limit statements. Continuity compares the value at a point with the limit there, and differentiability compares the difference quotient with a limiting slope. The algebra of limits lets us combine known continuous functions without repeating the definition. For a quotient the denominator must be non-zero, and for a composition the inner function must stay inside the outer domain.

1.3 Derivative rules you must know cold

The chain rule carries most of this chapter. Every derivative question that is not a plain formula is a chain rule question in disguise. Learn the derivatives of the inverse trigonometric functions, the exponentials and the logarithms, then practise spotting the inner function.

For a function raised to a function, take logarithms first. For a curve given by parameter t, divide dy by dt by dx by dt. For the second derivative of a parametric curve, differentiate dy by dx again with respect to t and divide by dx by dt.

Differentiability needs the left and right derivatives to agree, and continuity alone does not guarantee it. The absolute value function is the standard example of a continuous function that fails this test at a corner.