Which of the follwoing is always correct? (A) If f^'(x)>0 ∀x wherever f^'(x) exists, then f(x) m

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Which of the follwoing is always correct? (A) If f^'(x)>0 ∀x...

Which of the follwoing is always correct? (A) If $f^{\prime}(x)>0 \forall x$ wherever $f^{\prime}(x)$ exists, then $f(x)$ must be one-one (B) If $f^{\prime}(x)<0 \forall x$ wherever $f^{\prime}(x)$ exists, then $f(x)$ must be one-one (C) If $\mid f(x)$ be continuous at $x=a$, then $f(x)$ is also continuous at $\mathrm{x}=\mathrm{a}$ (D) If $f(x)$ is continuous at $x=a, f(A)=2$ and $x=a$ is the point of local minimum of $f(x)$, then $[f(x)]$, where [] denotes gresatest integer function, is also continuous

Which of the follwoing is always correct?
(A) If $f^{\prime}(x)>0 \forall x$ wherever $f^{\prime}(x)$ exists, then $f(x)$ must be one-one
(B) If $f^{\prime}(x)<0 \forall x$ wherever $f^{\prime}(x)$ exists, then $f(x)$ must be one-one
(C) If $\mid f(x)$ be continuous at $x=a$, then $f(x)$ is also continuous at $\mathrm{x}=\mathrm{a}$
(D) If $f(x)$ is continuous at $x=a, f(A)=2$ and $x=a$ is the point of local minimum of $f(x)$, then $[f(x)]$, where [] denotes gresatest integer function, is also continuous

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Key Concepts

Greatest Integer (Floor) Function

The greatest integer or floor function maps any real number to the largest integer less than or equal to that number. It is characterized by jump discontinuities at integer values, meaning that even if an underlying function is continuous, applying the floor function can introduce discontinuities, especially when the value passes through an integer.

Absolute Value Function

The absolute value function, which outputs the non-negative magnitude of a number, is continuous everywhere. However, while the absolute value of a function being continuous at a point implies certain regularities, it does not necessarily guarantee that the original function is continuous at that same point.

Continuity of Functions

A function is said to be continuous at a point if the limit of the function as it approaches that point is equal to the function’s value at that point. This concept is fundamental in ensuring that there are no jumps, breaks, or gaps in the function’s graph at that point.

Monotonicity and Injectivity

A function whose derivative is strictly positive (or strictly negative) wherever it exists is strictly monotonic (increasing or decreasing, respectively). A strictly monotonic function is one-to-one, meaning that each output is produced by exactly one input.

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