Exercise · ZW-CONTINUOUS-FUNCTION-008 · ★★★ Advanced
Which statement about the continuity of \(f\) at a point \(a\) is correct?
Choose one option.
Show answer
- B. If \(f\) is continuous at \(a\), then \(\lim_{x\to a}f(x)=f(a)\)
Explanation
By definition, continuity at \(a\) requires the limit to equal \(f(a)\). The others are false: \(|x|\) is continuous at 0 but not differentiable, the limit can exist with \(f(a)\) different, and there are removable discontinuities.
Review the concept: Continuous function
A function \(f\) is continuous at \(a\) if \(f(a)\) is defined, its limit as \(x\) tends to \(a\) exists and equals \(f(a)\).
\[\lim_{x\to a}f(x)=f(a)\]
More practice problems: continuous function
- ZW-CONTINUOUS-FUNCTION-001 Which of the following is NOT a condition for f to be continuous at a?
- ZW-CONTINUOUS-FUNCTION-002 At which point is the function f(x)=(1)/(x-4) discontinuous?
- ZW-CONTINUOUS-FUNCTION-003 Which value of k makes f(x)=begincases(x^2-4)/(x-2)&if x≠22mm] k&if x=2endcases continuous at x = 2?
- ZW-CONTINUOUS-FUNCTION-004 Which value of k makes f(x)=begincasesx+1&if x<1\\ kx^2&if x≥1endcases continuous at x = 1?
- ZW-CONTINUOUS-FUNCTION-005 Which value of k makes f(x)=begincaseskx-3&if x≤2\\ x^2-1&if x>2endcases continuous at x = 2?
- ZW-CONTINUOUS-FUNCTION-006 Which value of k makes f(x)=(x^2-9)/(x-3) continuous at x = 3 when x≠3 and f(3)=k?
- ZW-CONTINUOUS-FUNCTION-007 Which value of k makes continuous at x = 1 the function defined by f(x)=x+k if x<1 and f(x)=2x^2 if…
- ZW-CONTINUOUS-FUNCTION-009 What type of discontinuity does f(x)=(x^2-1)/(x-1) have at x = 1?