Practice question · Sort into groups
Sort each function by whether it is continuous at every real number.
Groups: Continuous everywhere · Not continuous everywhere
- (x + 1)/(x squared - 9)
- e to the power x
- the absolute value of x
- cos x
- 1/(x - 2)
- x cubed - 2x + 1
Hints
- Polynomials, sine, cosine and exponentials are continuous everywhere, and so are their sums and products.
- A quotient can only fail where its denominator vanishes.
Show the answer
Continuous everywhere: x cubed - 2x + 1, cos x, e to the power x, the absolute value of x
Not continuous everywhere: 1/(x - 2), (x + 1)/(x squared - 9)
Why
The two quotients break where their denominators vanish, at x = 2 and at x = 3 and -3. The absolute value is the interesting one: it has a sharp corner at 0 yet is perfectly continuous there, the corner will block DIFFERENTIABILITY in the next lesson, which is a strictly stronger demand.
Practise Continuity
The app has 5 more questions on this lesson, and keeps your place in the course. Mathematics I is free to start.
More questions on Continuity
- Continuity at a point requires three separate conditions, and the limit existing is only one of them.
- Order the checks that establish continuity of f at the point a.
- The function |x| is continuous everywhere and has no derivative at 0. What does that pair of facts establish…
- f is continuous on the closed interval from 0 to 4. Select every statement that MUST be true.