a) If a function f(x) is differentiable at x=t, it also has to be continuous at x=t.

b) If a function f(x) is continuous at x=t, it also has to be differentiable at x=t.

c) If a function f(x) is not differentiable at x=t, it cannot be continuous at x=t.

d) If a function f(x) is not continuous at x=t, it cannot be differentiable at x=t.

e) If a function f(x) is differentiable at x=t, then lim (x→t) f(x)=f(t).

f) If lim (x→t) f(x)=f(t), then f is differentiable at x=t.

# Which of the following statements are true? There can be multiple true statements; indicate all of them.

Share

Let’s analyze each statement one by one:

a) If a function f(x) is differentiable at x=t, it also has to be continuous at x=t.

This statement is true. According to the definition of differentiability, if a function is differentiable at a point, it must be continuous at that point. Differentiability implies continuity.

b) If a function f(x) is continuous at x=t, it also has to be differentiable at x=t.

This statement is not true. Continuous functions do not necessarily have to be differentiable. There exist continuous functions that are not differentiable at certain points.

c) If a function f(x) is not differentiable at x=t, it cannot be continuous at x=t.

This statement is not true. A function can be continuous at a point even if it is not differentiable at that point. Differentiability is a stronger condition than continuity.

d) If a function f(x) is not continuous at x=t, it cannot be differentiable at x=t.

This statement is true. If a function is not continuous at a point, it cannot be differentiable at that point. Differentiability requires continuity.

e) If a function f(x) is differentiable at x=t, then lim (x→t) f(x)=f(t).

This statement is true. If a function is differentiable at a point, then it is also continuous at that point. The limit of the function as x approaches t will be equal to the value of the function at t.

f) If lim (x→t) f(x)=f(t), then f is differentiable at x=t.

This statement is not true. The existence of the limit does not imply differentiability. A function can have a limit at a point without being differentiable at that point.

In summary, the true statements are:a) If a function f(x) is differentiable at x=t, it also has to be continuous at x=t.

d) If a function f(x) is not continuous at x=t, it cannot be differentiable at x=t.

e) If a function f(x) is differentiable at x=t, then lim (x→t) f(x)=f(t).