Continuous Function Math Example 1

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

Example 1

easy
Show that f(x)=3x2−5x+2f(x) = 3x^2 - 5x + 2 is continuous at x=1x = 1 using the three-part definition of continuity.

Solution

  1. 1
    Part 1 — f(1)f(1) exists: f(1)=3(1)−5(1)+2=0f(1) = 3(1)-5(1)+2 = 0. ✓
  2. 2
    Part 2 — lim⁡x→1f(x)\lim_{x\to1} f(x) exists: since ff is a polynomial, the limit equals the function value. lim⁡x→1(3x2−5x+2)=3−5+2=0\lim_{x\to1}(3x^2-5x+2) = 3-5+2 = 0. ✓
  3. 3
    Part 3 — Limit equals function value: lim⁡x→1f(x)=0=f(1)\lim_{x\to1}f(x) = 0 = f(1). ✓ All three conditions hold, so ff is continuous at x=1x=1.

Answer

ff is continuous at x=1x = 1
Continuity requires three conditions: the function value exists, the limit exists, and they are equal. Polynomials satisfy all three at every point, making them everywhere continuous.

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 2 hard

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

Example 3 easy

Is [formula] continuous on [formula]? If not, identify where and why.

Example 4 medium

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

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