Basic concept and visualisation of Continuity

Compulsory

Mathematics

Grade 11 11

TOOLS

Size 1 Size 2 Size 3 Size 4

chapter-17.4

Continuity

Solution by
Ghanashyam Adhikari

By Ghanashyam Adhikari

Activity 1

Curve Tracing & Limit Convergence

वक्र रेखा ट्रेसिङ र सीमा अभिसरण

A curve is continuous if it can be drawn without lifting the pen. Try dragging the tracer dots below. Notice that on Curve (ii), you encounter a jump gap that prevents smooth tracing!

यदि कुनै वक्र रेखा कलम नउठाई कोर्न सकिन्छ भने त्यो निरन्तर (Continuous) हुन्छ। तलका मार्करहरू ट्रेस गर्नुहोस्। रेखाचित्र (ii) मा विच्छेदन (gap) देखिनेछ!

Fig (i): Continuous Curve

1

Drag marker 1 across the curve!

Fig (ii): Discontinuous Curve (Jump)

2

Jump gap hit! Lift marker to proceed to the next segment.

X	1.9	1.99	1.999	1.9999	x → 2-
f(x)	10.60	10.96	10.996	10.9996	11.00

X	2.1	2.01	2.001	2.0001	x → 2+
f(x)	11.40	11.04	11.004	11.0004	11.00

Analysis and Continuity Condition:

Since LHL = RHL = Functional Value, the function is continuous at x = 2!

Textbook Worked Out Examples

Concept Assessment (Exercises)

Q No.

Mathematical Continuity Laws

1. Continuity Criteria

A function $f(x)$ is defined to be continuous at $x = a$ if and only if:

1. $f(a)$ is defined (returns a finite real number).
2. $\lim_{x \to a} f(x)$ exists.
3. The limit is equal to the functional value:

2. Core Theorems

If $f$ and $g$ are real-valued continuous functions defined in the neighborhood of $a$, then:

A $cf$ is continuous at $a$ (for any real constant $c$).

B $f + g$ and $f - g$ are continuous at $x = a$.

C $f \cdot g$ is continuous at $x = a$.

D $\frac{f}{g}$ is continuous at $x = a$, provided $g(a) \neq 0$.

Types of Discontinuity: Graph and Example Explorer

Continuous Function

Interactive Canvas

Variable target position ($x$): 1.00

$x \to a^-$ (Left Limit) $x \to a^+$ (Right Limit)

Theoretical Condition

LHL ($x \to a^-$)

-

RHL ($x \to a^+$)

-

Value $f(a)$

-

Matched Case Proof

Analysis Verdict

Cartesian Graph Sandbox Plotter

Define Sandbox Equation

Choose Function Preset: Step Function: f(x) = [x] Rational Function: f(x) = (x² - 1)/(x - 1) Reciprocal function: f(x) = 1/x Dampened Oscillating: f(x) = x * sin(1/x)

Active Expression:

Tip: Enable "PEN" from the top toolbar to sketch directly over this sandbox canvas graph.