30 Questions
What can be done to remove a discontinuity of a function f at x = a?
Assign a suitable value to the function f at x = a
What can be said about the continuity of f + g and f - g if f is continuous at x = c and g is discontinuous at x = c?
f + g and f - g are discontinuous
What is the point of discontinuity of the function y = u + u - 2 where u = -1?
x = -1
What is true about the function f(x) = 1/(x - 1)?
It is discontinuous at x = 1
What is the domain of the function log(1 + ax) - log(1 - bx)?
All real numbers except x = 0
What is the meaning of the word 'Continuous' in mathematics?
Without any break or gap
Which of the following functions is NOT continuous?
[x]
A function f(x) is said to be continuous at a point x=a if and only if:
f(a) exists and lim f(x) = f(a)
Which of the following functions is continuous on the entire real line?
sin x
What is the condition for a function f(x) to be discontinuous at a point x=a?
f(a) exists and lim f(x) ≠ f(a)
If a continuous function f(x) is defined on [a, b] and f(a) and f(b) are of opposite signs, then what can be concluded about the function?
There is at least one value of x for which f(x) vanishes.
If a function f(x) is continuous on [a, b] and maps [a, b] into [a, b], what can be concluded about the function?
For some x in [a, b], we have f(x) = x.
What is a necessary condition for the composite function y = (gof)(x) = g(f(x)) to be continuous at x = a?
The function f(x) is continuous at x = a and g(u) is continuous at u = f(a).
What type of discontinuity occurs when lim+ f(x) = lim- f(x) but their common value is not equal to f(a)?
Removable discontinuity
What is the condition for a function f to be discontinuous at x = a?
lim+ f(x) or lim- f(x) or both may not exist.
Which of the following functions is continuous on the entire real line?
e^x
What can be said about the domain of the function f(x) = 1/(x^2 + 1)?
It is defined on the entire real line.
Let f(x) and g(x) be two continuous functions at x = a. Which of the following is continuous at x = a?
All of the above
What can be said about the function f(x) = x^2 at x = 0?
It is continuous and differentiable.
Let f(x) be a continuous function on [a, b]. What can be concluded about the function f(x)?
It is bounded on the interval [a, b].
What can be said about the function f(x) = 2|x|, in terms of its continuity?
It is continuous on the entire real line.
What is the domain of the function f(x) = 1/(sin(x) - 1), if it is continuous?
All real numbers except x = nπ, where n is an integer
Which of the following functions has a discontinuity at x = 0?
f(x) = 1/x
What can be said about the function f(x) = cos(x), in terms of its continuity?
It is continuous on the entire real line.
What is true about the function f(x) = |x|, in terms of its continuity?
It is continuous on the entire real line, but not differentiable at x = 0.
What is the domain of the function f(x) = |x - 2|?
(-∞, ∞)
If f(x) = x^2 and g(x) = 1/x, then what is the domain of the composite function (fog)(x)?
(-∞, 0) U (0, ∞)
Which of the following functions is continuous on the entire real line?
f(x) = 1/(x^2 + 1)
If f(x) = x^2 is continuous on [0, 1] and f(0) = 0 and f(1) = 1, then what can be concluded about the function?
The function has a root in [0, 1]
If f(x) is discontinuous at x = a, then what can be said about the limits of f(x) at x = a?
lim+ f(x) = lim- f(x) but not equal to f(a)
Test your understanding of functions, limits, and continuity with this quiz. Learn about removable and non-removable discontinuities, and how they affect functions. Practice your problem-solving skills with these challenging questions.
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