Continuity

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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 (correct)

Remove the function f from the domain

Replace the function f with a different function

Change the value of x to a different value

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 (correct)

Both f + g and f - g are continuous

Both f + g and f - g are discontinuous

f + g is continuous, but f - g is discontinuous

What is the point of discontinuity of the function y = u + u - 2 where u = -1?

x = -2

x = 2

x = 1

x = -1 (correct)

What is true about the function f(x) = 1/(x - 1)?

It is discontinuous at x = 1 Signup and view all the answers

What is the domain of the function log(1 + ax) - log(1 - bx)?

All real numbers except x = 0 Signup and view all the answers

What is the meaning of the word 'Continuous' in mathematics?

Without any break or gap Signup and view all the answers

Which of the following functions is NOT continuous?

[x] Signup and view all the answers

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) Signup and view all the answers

Which of the following functions is continuous on the entire real line?

sin x Signup and view all the answers

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) Signup and view all the answers

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. Signup and view all the answers

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. Signup and view all the answers

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). Signup and view all the answers

What type of discontinuity occurs when lim+ f(x) = lim- f(x) but their common value is not equal to f(a)?

Removable discontinuity Signup and view all the answers

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. Signup and view all the answers

Which of the following functions is continuous on the entire real line?

e^x Signup and view all the answers

What can be said about the domain of the function f(x) = 1/(x^2 + 1)?

It is defined on the entire real line. Signup and view all the answers

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 Signup and view all the answers

What can be said about the function f(x) = x^2 at x = 0?

It is continuous and differentiable. Signup and view all the answers

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]. Signup and view all the answers

What can be said about the function f(x) = 2|x|, in terms of its continuity?

It is continuous on the entire real line. Signup and view all the answers

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 Signup and view all the answers

Which of the following functions has a discontinuity at x = 0?

f(x) = 1/x Signup and view all the answers

What can be said about the function f(x) = cos(x), in terms of its continuity?

It is continuous on the entire real line. Signup and view all the answers

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. Signup and view all the answers

What is the domain of the function f(x) = |x - 2|?

(-∞, ∞) Signup and view all the answers

If f(x) = x^2 and g(x) = 1/x, then what is the domain of the composite function (fog)(x)?

(-∞, 0) U (0, ∞) Signup and view all the answers

Which of the following functions is continuous on the entire real line?

f(x) = 1/(x^2 + 1) Signup and view all the answers

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] Signup and view all the answers

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) Signup and view all the answers

Continuity

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Questions and Answers

What can be done to remove a discontinuity of a 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?

What is the point of discontinuity of the function y = u + u - 2 where u = -1?

What is true about the function f(x) = 1/(x - 1)?

What is the domain of the function log(1 + ax) - log(1 - bx)?

What is the meaning of the word 'Continuous' in mathematics?

Which of the following functions is NOT continuous?

A function f(x) is said to be continuous at a point x=a if and only if:

Which of the following functions is continuous on the entire real line?

What is the condition for a function f(x) to be discontinuous at a point x=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?

If a function f(x) is continuous on [a, b] and maps [a, b] into [a, b], what can be concluded about the function?

What is a necessary condition for the composite function y = (gof)(x) = g(f(x)) to be continuous at x = a?

What type of discontinuity occurs when lim+ f(x) = lim- f(x) but their common value is not equal to f(a)?

What is the condition for a function f to be discontinuous at x = a?

Which of the following functions is continuous on the entire real line?

What can be said about the domain of the function f(x) = 1/(x^2 + 1)?

Let f(x) and g(x) be two continuous functions at x = a. Which of the following is continuous at x = a?

What can be said about the function f(x) = x^2 at x = 0?

Let f(x) be a continuous function on [a, b]. What can be concluded about the function f(x)?

What can be said about the function f(x) = 2|x|, in terms of its continuity?

What is the domain of the function f(x) = 1/(sin(x) - 1), if it is continuous?

Which of the following functions has a discontinuity at x = 0?

What can be said about the function f(x) = cos(x), in terms of its continuity?

What is true about the function f(x) = |x|, in terms of its continuity?

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)?

Which of the following functions is continuous on the entire real line?

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?

If f(x) is discontinuous at x = a, then what can be said about the limits of f(x) at x = a?

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