Find all the points of discontinuity of the function $$f$$ defined by $$f(x)=\left\{\begin{array}{rr} x+2, & \text { if } x<1 0, & \text { if } x=1 x-2, & \text { if } x>1 \end{array}\right.$$

Step 1: Analyze continuity for $$x < 1$$

For all values of $$x < 1$$ , the function is defined as $$f(x) = x+2$$ . This is a polynomial function, and polynomial functions are continuous everywhere. Therefore, f(x) is continuous for all $$x < 1$$ .

Step 2: Analyze continuity for $$x > 1$$

For all values of $$x > 1$$ , the function is defined as $$f(x) = x-2$$ . This is also a polynomial function, and thus it is continuous for all $$x > 1$$ .

Step 3: Check continuity at $$x = 1$$

To check for continuity at $$x=1$$ , we need to evaluate the left-hand limit, the right-hand limit, and the function value at $$x=1$$ . The left-hand limit is $$\lim_{x \to 1^-} (x+2) = 3$$ . The right-hand limit is $$\lim_{x \to 1^+} (x-2) = -1$$ . The function value at $$x=1$$ is given as $$f(1) = 0$$ .

Step 4: Determine discontinuity at $$x = 1$$

Since the left-hand limit (3) is not equal to the right-hand limit (-1), the limit of the function as $$x$$ approaches 1 does not exist. For a function to be continuous at a point, the limit must exist and be equal to the function's value at that point. Therefore, the function is discontinuous at $$x=1$$ .

More from this chapter

• 2. Consider the function f(x) = 3cos(e^(5x-3)) + (4)/(x) Find f'(x) [(d)/(dx)(cos x) = -sin x]
• **प्रश्नावली 5.2** प्रश्न 1 से 8 में x के सापेक्ष निम्नलिखित फलनों का अवकलन कीजिए: 1. sin(x² + 5) 2. cos(sin x) 3. sin(ax + b) 4. sec(tan(√x)) 5. (sin(ax + b))/
• 1. Prove that the function f(x) = 2x² - 3x + 2 is continuous at x = 1. 2. The function f(x) is defined as follows: f(x) = 2x - 3, & x < 2 x - 1, & x ≥ 2 Prove t
• Find dy/dx if y = (ln(x))^(cos(x)).
• Differentiate the following w.r.t. x. (i) cos^-1(sin x) (ii) tan^-1(sin x / (1 + cos x)) (iii) sin^-1(2^(x+1) / (1 + 4^x))
• If x = t² and y = t³, then (d² y)/(dx²) is equal to:
• Find (d y)/(d x), if x=a(θ+sin θ), y=a(1-cos θ).
• Differentiate the following w.r.t. x. (i) cos ^(-1)(sin x) (ii) tan ^(-1)((sin x)/(1+cos x)) (iii) sin ^(-1)((2^(x+1))/(1+4^x))

All Continuity and Differentiability solutions →