Maths with David

IBDP. AA. HL. Differentiation. Continuity

Definition of “Continuous”

A function, f, is continuous at x = a if f(a) and $lim_{x \to a} f(x)$ exist and are equal.

Alternatively, we can say that f is continuous at x=a if at least one of the following is true:
- a is the left endpoint of an interval on which f is defined and f(a) and $lim_{ x \to a^{+} } f(x)$ exist and are equal;
- a is the right endpoint of an interval on which f is defined and f(a) and $lim_{ x \to a^{-} } f(x)$ exist and are equal.

If f is not continuous at x=a, we say either that f is discontinous at x = a, or that f has a discontinuity at x = a.

Regarding discontinuities, we can distinguish between a hole, a jump and a break.

A hole refers to a single undefined point in a continuous graph. A jump is similar, but involves the right hand side and left hand side of the jump not being connected. A break occurs when the function to the left of the break differs from the function to the right of the break. These subtle distinctions are demonstrated in the diagram below.

A hole is a removable discontinuity, which can be removed by defining a new function f whose value at x=a is defined as $lim_{x \to a} f(x)$ . Jumps and breaks are essential discontinuities, which cannot be removed by simply redefining the value of the function at those points.

Worked Example

Discuss the continuity of the following functions. If there is a removable discontinuity, describe how it could be removed: