Master continuity of functions for CBSE Class 11 Applied Mathematics. Learn limits, point continuity, and jump discontinuities with solved examples.
A function f(x) is said to be continuous at a point x = c if the limit of the function as x approaches c exists and is equal to the value of the function at that point. Mathematically, this requires the left-hand limit, right-hand limit, and the function value to be identical. If any of these conditions fail, the function is discontinuous at x = c.
Step 1: Find f(1) = 2(1) + 3 = 5.
Step 2: Find lim(x→1) (2x + 3) = 2(1) + 3 = 5.
Step 3: Since lim(x→1) f(x) = f(1), the function is continuous at x = 1.
Polynomial functions are defined as expressions involving powers of x with constant coefficients. A fundamental property in calculus is that every polynomial function is continuous for all real numbers. This simplifies the analysis of functions like f(x) = ax² + bx + c.
Step 1: Calculate f(2) = 2² - 4 = 0.
Step 2: Calculate lim(x→2) (x² - 4) = 2² - 4 = 0.
Step 3: Since limit equals function value, f(x) is continuous at x = 2.
Piecewise functions change their definition at specific points, often leading to potential discontinuities. To check for continuity, we must evaluate the limits from both sides of the transition point. If the left-hand limit does not equal the right-hand limit, the function has a jump discontinuity.
Step 1: LHL = lim(x→2⁻) (x+1) = 3.
Step 2: RHL = lim(x→2⁺) (3x-2) = 3(2)-2 = 4.
Step 3: Since 3 ≠ 4, the function is discontinuous at x = 2.