A function

*f*(

*x*) is said to be

**continuous** at a point (

*c*,

*f*(

*c*)) if each of the following conditions is satisfied:

Geometrically, this means that there is no gap, split, or missing point for *f*( *x*) at *c* and that a pencil could be moved along the graph of *f*( *x*) through ( *c*, *f*( *c*)) without lifting it off the graph. A function is said to be continuous at ( *c*, *f*( *c*)) from the right if and continuous at ( *c*, *f*( *c*)) from the left if . Many of our familiar functions such as linear, quadratic and other polynomial functions, rational functions, and the trigonometric functions are continuous at each point in their domain.

A special function that is often used to illustrate one‐sided limits is the greatest integer function. The *greatest integer function,* [ *x*], is defined to be the largest integer less than or equal to *x* (see Figure 1).

**Figure 1 **The graph of the greatest integer function *y* = [ *x*].

Some values of [ *x*] for specific *x* values are

The greatest integer function is continuous at any integer *n* from the right only because

hence, and *f*( *x*) is not continuous at *n* from the left. Note that the greatest integer function is continuous from the right and from the left at any noninteger value of *x*.

**Example 1:** Discuss the continuity of *f*( *x*) = 2 *x* + 3 at *x* = −4.

When the definition of continuity is applied to *f*( *x*) at *x* = −4, you find that

hence, *f* is continous at *x* = −4.

**Example 2:** Discuss the continuity of

When the definition of continuity is applied to *f*( *x*) at *x* = 2, you find that *f*(2) does not exist; hence, *f* is not continuous (discontinuous) at *x* = 2.

**Example 3:** Discuss the continuity of

When the definition of continuity is applied to *f*( *x*) at *x* = 2, you find that

hence, *f* is continous at *x* = 2.

**Example 4:** Discuss the continuity of .

When the definition of continuity is applied to *f*( *x*) at *x* = 0, you find that

hence, *f* is continuous at *x* = 0 from the right only.

**Example 5:** Discuss the continuity of

When the definition of continuity is applied to *f*( *x*) at *x* = −3, you find that

Many theorems in calculus require that functions be continuous on intervals of real numbers. A function *f*( *x*) is said to be continuous on an open interval ( *a*, *b*) if *f* is continuous at each point *c* ∈ ( *a*, *b*). A function *f*( *x*) is said to be continuous on a closed interval [ *a*, *b*] if *f* is continuous at each point *c* ∈ ( *a*, *b*) and if *f* is continuous at *a* from the right and continuous at *b* from the left.

**Example 6:**

a. *f*( *x*) = 2 *x* + 3 is continuous on (−∞,+∞) because *f* is continuous at every point *c* ∈ (−∞,+∞).

b*. f*( *x*) = ( *x* − 3)/( *x* + 4) is continuous on (−∞,−4) and (−4,+∞) because *f* is continuous at every point *c* ∈ (−∞,−4) and *c* ∈ (−4,+∞)

c*. f*( *x*) = ( *x* − 3)/( *x* + 4) is not continuous on (−∞,−4] or [−4,+∞) because *f* is not continuous on −4 from the left or from the right.

d. is continuous on [0, +∞) because *f* is continuous at every point *c* ∈ (0,+∞) and is continuous at 0 from the right.

e*. f*( *x*) = cos *x* is continuous on (−∞,+∞) because *f* is continuous at every point *c* ∈ (−∞,+∞).

f*. f*( *x*) = tan *x* is continuous on (0,π/2) because *f* is continuous at every point *c* ∈ (0,π/2).

g*. f*( *x*) = tan *x* is not continuous on [0,π/2] because *f* is not continuous at π/2 from the left.

h*. f*( *x*) = tan *x* is continuous on [0,π/2) because *f* is continuous at every point *c* ∈ (0,π/2) and is continuous at 0 from the right.

i*. f*( *x*) = 2 *x*/( *x* ^{2} + 5) is continuous on (−∞,+∞) because *f* is continuous at every point *c* ∈ (−∞,+∞).

j*. f*( *x*) = | *x* − 2|/( *x* − 2) is continuous on (−∞,2) and (2,+∞) because *f* is continuous at every point *c* ∈ (−∞,2) and *c* ∈ (2,+∞).

k*. f*( *x*) = | *x* − 2|/( *x* − 2) is not continuous on (−∞,2] or [2,+∞) because *f* is not continuous at 2 from the left or from the right.