Which of the following statements is true?
- A continuous function is an increasing function.
- An increasing function is continuous.
- A continuous function is differentiable.
- A differentiable function is continuous.
The Correct Option is D
Solution and Explanation
A differentiable function is continuous, this statement is true because if a function is differentiable at a point, it implies that it is also continuous at that point. However, the converse is not necessarily true; a continuous function may not be differentiable at every point.
So, the correct option is (D): A differentiable function is continuous.
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Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.