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Chapter 5 · Continuity and Differentiability

Example 5: The constant function f(x)=kf(x)=k is continuous everywhere

ExamplesContinuous functions · Section 5.2
The question

Check the points where the constant function f(x)=kf(x) = k is continuous.

Exam answer

Let cc be any real number. Then f(c)=kf(c) = k⁠.

LHL=lim⁡x→c−k=k\text{LHL} = \lim_{x\to c^-} k = k RHL=lim⁡x→c+k=k\text{RHL} = \lim_{x\to c^+} k = k

Since LHL=RHL=f(c)\text{LHL} = \text{RHL} = f(c)⁠, ff is continuous at x=cx = c⁠.

As cc is arbitrary, f(x)=kf(x) = k is continuous at every real number.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

The question asks about every real number at once, so we test continuity at an arbitrary point cc⁠, a letter that could stand for any real number. For a constant function the test is almost immediate: every value, near cc or at cc⁠, is the same number kk⁠, so the limits and the value cannot disagree.

Given, to find, and the plan
Given
f(x)=kf(x) = k for every real xx⁠, where kk is a fixed real number.
To find
All points at which ff is continuous.
Plan
Fix an arbitrary real number cc⁠. Find f(c)f(c)⁠, the left-hand limit and the right-hand limit at cc⁠, and compare them (Definition 1). Since cc was arbitrary, the conclusion holds at every real number.

Basics you need

1What a constant function is

f(x)=kf(x) = k gives the same output kk for every input. Its graph is the horizontal line y=ky = k⁠. For example, f(x)=−4f(x) = -4 sends 00⁠, 22 and 100100 all to −4-4⁠. Basics: functions

2What a limit is

lim⁡x→cf(x)=L\lim_{x\to c} f(x) = L means f(x)f(x) can be made as close to LL as we like by taking xx close enough to cc⁠, x≠cx \ne c⁠. Basics: what a limit is

3Proving a "for every" statement

Work with a letter cc about which we assume nothing except that it is a real number. Whatever we prove about cc then holds for every real number.

4Definitions 1 and 2

Definition 1: ff is continuous at cc (in the domain) iff the left-hand limit, right-hand limit and f(c)f(c) exist and are equal. Definition 2: ff is a continuous function if it is continuous at every point of its domain. Lesson: continuity at a point

Solution

  1. Fix an arbitrary point and find f(c)f(c)

    The rule f(x)=kf(x) = k gives a value for every real xx⁠, so the domain is R\R⁠. Let cc be any real number. Then

    f(c)=k.f(c) = k.
    The question asks "at which points?", and the answer could be "all", "none" or "some". Working with an arbitrary cc covers every candidate at once (Basic 3). The domain check comes first because continuity is only discussed at points of the domain.
    "By definition, its value at any real number equals kk" (the textbook's words): that is what "constant" means. Basics: functions

  2. Left-hand limit at cc

    For every x<cx < c⁠, f(x)=kf(x) = k⁠. So

    lim⁡x→c−f(x)=lim⁡x→c−k=k.\lim_{x\to c^-} f(x) = \lim_{x\to c^-} k = k.
    The values of ff to the left of cc are not just close to kk⁠, they are kk⁠: the distance ∣f(x)−k∣|f(x) - k| is exactly 0 for every xx⁠. So whatever closeness is demanded, it is met, and the limit is kk⁠.
    The limit of a constant is that constant. Basics: one-sided limits

  3. Right-hand limit at cc

    For every x>cx > c⁠, again f(x)=kf(x) = k⁠, so

    lim⁡x→c+f(x)=lim⁡x→c+k=k.\lim_{x\to c^+} f(x) = \lim_{x\to c^+} k = k.

    With k=−4k = -4 and c=2c = 2⁠, for instance:

    xxf(x)f(x)side of 2
    1.91.9−4-4left
    1.991.99−4-4left
    2.012.01−4-4right
    2.12.1−4-4right
    Both one-sided limits are kk⁠, so the limit at cc exists and lim⁡x→cf(x)=k\lim_{x\to c} f(x) = k⁠, which is the single line in the textbook's solution.

  4. Compare at cc

    lim⁡x→c−f(x)=k,lim⁡x→c+f(x)=k,f(c)=k.\begin{aligned} \lim_{x\to c^-} f(x) &= k, \\ \lim_{x\to c^+} f(x) &= k, \\ f(c) &= k. \end{aligned}

    All three agree, so lim⁡x→cf(x)=f(c)\lim_{x\to c} f(x) = f(c) and ff is continuous at cc (Definition 1).

    Definition 1 asks for exactly this equality. Nothing in the argument used any special property of cc⁠.
    Lesson: continuity at a point

  5. Conclude for every real number

    Since cc was an arbitrary real number, f(x)=kf(x) = k is continuous at every real number. As its domain is R\R⁠, it is a continuous function in the sense of Definition 2.

    Step 1 promised that an argument about an arbitrary cc proves the statement for all real numbers; Step 4 completed that argument.
    This result, together with Example 6 (the identity function), is the starting point for proving that every polynomial is continuous (Example 14).
x y O c k (c, k) y = k (drawn for k > 0) x → c from the left x → c from the right the value f(c) = k
A constant function is a horizontal line y=ky = k (drawn here with k>0k > 0⁠). Approaching any point cc from either side, the height never changes, and the point (c,k)(c, k) is on the line.
Result

For every real cc⁠: lim⁡x→cf(x)=k=f(c)\lim_{x\to c} f(x) = k = f(c)⁠. So the constant function f(x)=kf(x) = k is continuous at every real number, which is the textbook's conclusion.

Check it with numbers

Take k=−4k = -4 and test at c=2c = 2⁠: f(1.999)=−4f(1.999) = -4⁠, f(2.001)=−4f(2.001) = -4⁠, f(2)=−4f(2) = -4⁠. The distance from f(2)f(2) is 0 on both sides, at every distance from 2.

Common mistakes
  1. Testing a few points only. Showing continuity at 0, 1 and 2 does not show it at every real number. Use an arbitrary cc⁠.
  2. Thinking a limit must "approach" a different number. A function can equal its limit everywhere near cc⁠; the limit is still kk⁠.
  3. Forgetting the domain. Continuity is claimed at every point of the domain, here all of R\R⁠.
What to take away
  1. To prove continuity at every point, prove it at an arbitrary point cc⁠.
  2. Constant functions are continuous everywhere: ∣f(x)−f(c)∣=0|f(x) - f(c)| = 0 for all xx⁠.
  3. Definition 2: a function continuous at every point of its domain is called a continuous function.

Every fact used, and where it comes from

FactUsed inSource
A constant function takes the value kk everywhereSteps 1–3Definition of a constant function (Basics: functions)
lim⁡x→ck=k\lim_{x\to c} k = k (from either side)Steps 2, 3Algebra of limits, Class XI (Limits and Derivatives) (Basics: limits)
Continuous at cc iff LHL = RHL = f(c)f(c)Step 4Definition 1, Section 5.2 (Lesson: continuity at a point)
Continuous function: continuous at every point of the domainStep 5Definition 2, Section 5.2
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