Observe the graph of a function \( y=f(x) \) and identify the correct statements (s):
A. Function is continuous but not differentiable at \( x=a \)
B. Function is not defined at \( x=a \)
C. Function is defined at \( x=a \) but not differentiable.
D. Given that at \( x=a,y=b \) .
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → C only
In the graph, the curve approaches height b from both sides of x = a (open circle), but the value at x = a is c (filled dot):
\( \displaystyle \lim_{x\to a}f(x)=b \)
\( \displaystyle f(a)=c\ne b \)
So f is defined at a with f(a) = c, the limit there is b, and f is not continuous (hence not differentiable) at x = a.
Checking the statements
(A) Function is continuous but not differentiable at \( x=a \) — Incorrect. The limit b is not equal to f(a) = c, so f is not continuous at a.
(B) Function is not defined at \( x=a \) — Incorrect. The filled dot shows f(a) = c.
(C) Function is defined at \( x=a \) but not differentiable. — Correct. f(a) = c exists, and a function that is not continuous at a point is not differentiable there.
(D) Given that at \( x=a,y=b \) . — Incorrect. At x = a, y = c; b is only the limit.