Continuity and Discontinuity Hien Pham and Kim Lai

Importance Calculus relies heavily upon the existence of what’s called continuous functions. In fact, most of the important theorems require that any function in question must be continuous before one can even think about applying the theorems.

“Given a continuous function f(x)…” Continuity “As long as g(x) is a continuous function…” “Assume h(x) is continuous on [a,b]…” “Given a continuous function f(x)…”

Continuous functions are predictable… No breaks in the graph No holes No jumps

A function such as g(x)= A function such as g(x)= has a discontinuity at x=3 because the denominator is zero there

It is reasonable to say that the function is “continuous” everywhere else because the graph seems to have no other “gaps” or “jumps”

Continuity Continuity at a Point: A function f(x) is said to be continuous x = c if each of the following conditions is satisfied: f(c) exists, exists, and = f(c)

Continuity Continuity on an Interval: A function f(x) is continuous on an interval of x-values if and only if it is continuous at each value of x in that interval At the end points of a closed interval, only the one-sided limits need to equal the function

Cusp A cusp is a point on the graph at which the function is continuous but the derivative is discontinuous Verbally: A cusp is a sharp point or an abrupt change in direction

The graph can have a cusp (an abrupt change in direction) at x=c and still be continuous there

Removable Discontinuities f(x) = L exists (and is finite) but f(c) is not defined or f(c) ≠ L You can define or redefine the value of f(c) to make f continuous at this point

If f(x) = L but f(c) is not defined then the discontinuity at x=c can be removed by defining f(c) = L

We can “remove” the discontinuity by filling the hole Example Consider the function g(x) = . Then g(x) = (x + 1) for all real numbers except x=1 Since g(x) and x+1 agree all points other than the objective, We can “remove” the discontinuity by filling the hole The domain of g(x) may be extended to include x=1 by declaring that g(1)=2. This makes g(x) continuous at x=1. Since g(x) is continuous at all other points by defining g(x)=2 turns g into a continuous function.

If but f(a) ≠ L then the discontinuity at x=a can be removed by redefining f(a)=L

In this case, we redefine h(0.5)=1.5 + (1/.75) = 17/6 Example Unless 0<x<1 Consider the function If x=0.5 0<x<1, x≠0.5 We can remove the discontinuity by redefining the function so as to fill the hole In this case, we redefine h(0.5)=1.5 + (1/.75) = 17/6

You cannot remove a step discontinuity simply by redefining f(c) Although there is a value for f(c), f(x) approaches different values from the left of c and the right of c. So, there is no limit of f(x) as x approaches c. You cannot remove a step discontinuity simply by redefining f(c)

Example

Infinite Discontinuity As x gets closer to c, the value of f(x) becomes large without bound. The discontinuity is not removable just by redefining f(c)

The graph approaches a vertical asymptote at x = c Example The graph approaches a vertical asymptote at x = c

One-sided Limits and Piecewise Functions The graph is an example of a function that has different one-sided limits as x approaches c. -As x approaches c from the left side, f(x) stays close to 4. -As x approaches c from the right side, f(x) stays close to 7.

One-sided Limits x  c from the left (through values of x on the negative side of c) x  c from the right (through values of x on the positive side of c) if and only if and

Piecewise Function A step discontinuity can result if f(x) is defined by a different rule for c than it is for the piece to the left if x ≤ 2 if 2 < x < 5 if x ≥ 5 Each part of the function is called a branch. You can plot the three branches on your grapher by entering the three equations, then dividing by the appropriate Boolean variable. A Boolean variable, such as (x ≤ 2), equals 1 if the condition inside the parenthesis is true and 0 if the condition is false.

Example 1 For the piecewise function f shown, if x ≤ 2 For the piecewise function f shown, a. Does f(x) have a limit as x approaches 2? Explain. Is f continuous at x = 2? b. Does f(x) have a limit as x approaches 5? Explain. Is f continuous at x = 5? if 2 < x < 5 if x ≥ 5

The function f is discontinuous at x = 2. Solution (part a) and The left and right limits are unequal. does not exist. There is a step discontinuity. The function f is discontinuous at x = 2.

Solution (part b) and The left and right limits are equal. The open circle at the right end of the middle branch is filled with the closed dot on the left end of the right branch. The function f is continuous at x = 5 because the limit as x approaches 5 is equal to the function value at 5.

Example 2 Let the function if x < 2 Let the function if x ≥ 2 Find the value of k that makes the function continuous at x = 2. Plot and sketch the graph.

For h to be continuous at x = 2, the two limits must be equal. Solution (part a) For h to be continuous at x = 2, the two limits must be equal.

Solution (part b) The missing point at the end of the left branch is filled by the point at the end of the right branch, showing graphically that h is continuous at x = 2.

The End