Continuity

The last lesson promised a name for the easy, well-behaved case: a curve that flows straight through its point, with no hole and no jump. Here it is, in the plainest English there is — the graph you can draw without lifting your pen off the paper. No jumps, no holes, no shooting off to infinity; the curve just keeps going.

That unbroken quality is continuity, and it is the property a loss function needs before gradient descent can trust it — a jump or a hole is exactly where an optimiser stumbles, which is why continuity quietly underpins most of machine learning. To pin the idea down, we lean on the limit machinery from the last three lessons. At a point a, the curve must arrive at exactly the value the function takes there: the value it approaches and the value it actually holds have to be one and the same number.

That single demand secretly bundles three. f(a) must exist, the limit as x → a must exist, and the two must agree. Miss any one of the three and you have a discontinuity — and GATE’s favourite version of this question hands you a piecewise function and asks for the constant that makes the pieces meet.

Continuity at a point

f is continuous at a when:

lim  f(x)  =  f(a)
x→a

This one line silently demands all three conditions at once: f(a) exists, the two-sided limit exists, and they are equal. Stretch the idea across a whole stretch of the axis and f is continuous on an interval when it is continuous at every point of that interval.

Kinds of discontinuity

When the three conditions fail, they fail in one of three recognisable ways:

• Removable (a hole) — the two-sided limit exists, but f(a) is missing or set to a different value. You could “fill the hole” and fix it in one stroke.
• Jump — the left-hand and right-hand limits both exist but disagree, so the two-sided limit does not exist (the right-hand picture above). No single f(a) can repair it.
• Infinite — the function blows up near a (a vertical asymptote), as 1/x does at x = 0.

Matching a piecewise function

Now the case GATE actually tests. When f is built from different formulas on either side of a point a, continuity at that seam forces the pieces to meet: the left formula and the right formula must hand back the same value at a. That single equality is usually the whole question — set the two expressions equal at the seam and solve for the unknown constant.

How GATE asks this

The classic format is MCQ or NAT: a piecewise function carrying an unknown constant k (or a), with the instruction “find k so that f is continuous.” You write the seam-matching equation — left piece at a equals right piece at a — and solve. GATE DA 2024 posed exactly this: a two-piece function joined at a point, cracked by equating the pieces there. Other variants ask you instead to classify a discontinuity, or to pick which point breaks continuity.

Worked example — solve for the constant (GATE DA 2024 style)

Find the constant k that makes the following continuous at x = 2:

f(x) = x + 1 for x ≤ 2, and f(x) = k·x for x > 2.

Continuity at the seam x = 2 needs the limit from the right to match the value from the left — that is, the two pieces must agree at x = 2:

left piece at 2 :  f(2) = 2 + 1 = 3
right limit at 2:  lim_{x→2⁺} k·x = k·2 = 2k

match:   2 + 1 = k · 2     →     3 = 2k     →     k = 3/2 = 1.5

So k = 1.5. With that value both pieces hit 3 at x = 2, the graph connects, and f is continuous there. Any other k leaves a jump — and the guess you locked in tells you whether your instinct for the seam was already sound.

A question to carry forward

Continuity buys you a curve with no breaks — you can trace it without lifting the pen. But recall the V-shape |x| from two lessons back: you can draw that without lifting the pen too, so it is perfectly continuous, and yet it carries a sharp corner at x = 0 where the slope can’t decide between −1 and +1. So “no break” plainly is not the end of the story about smoothness. Here is the thread onward: is there a stronger kind of well-behavedness that forbids corners as well as breaks — one that pins down a single, definite slope at every point?

In one breath

• Continuous at a means lim_{x→a} f(x) = f(a) — one line bundling three demands: f(a) exists, the limit exists, and the two are equal.
• Continuous on an interval = continuous at every point of it; in plain terms, drawable without lifting the pen.
• Three discontinuities: removable (hole — limit fine, value wrong/missing), jump (one-sided limits disagree), infinite (blows up, like 1/x at 0).
• Piecewise seam: set the left and right formulas equal at a and solve for the constant — GATE DA 2024 was exactly this (x+1 = kx at 2 ⇒ k = 1.5).
• The trap: a limit existing is not enough — it must also equal f(a).

Practice