Continuous Function Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Is g(x)=1xg(x) = \dfrac{1}{x} continuous on (−∞,∞)(-\infty, \infty)? If not, identify where and why.

Solution

  1. 1
    g(x)=1/xg(x) = 1/x is undefined at x=0x = 0 (division by zero), so the first condition of continuity fails there.
  2. 2
    Therefore gg is not continuous on all of R\mathbb{R}; it is discontinuous at x=0x = 0 with an infinite discontinuity (vertical asymptote).

Answer

Not continuous on (−∞,∞)(-\infty,\infty); discontinuous at x=0x=0 (infinite discontinuity)
A function cannot be continuous at a point where it is undefined. The reciprocal function has a vertical asymptote at x=0x=0, which is an infinite (essential) discontinuity — it cannot be removed by redefining the function value.

About Continuous Function

A function is continuous at a point if the limit equals the function value there, with no jumps, holes, or vertical asymptotes in the interval of interest.

Learn more about Continuous Function →

More Continuous Function Examples

Example 1 easy

Show that [formula] is continuous at [formula] using the three-part definition of continuity.

Example 2 hard

Find where [formula] is discontinuous, classify the discontinuity, and determine if it can be remove

Example 4 medium

Apply the Intermediate Value Theorem to show that [formula] has a root in the interval [formula].

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